maths.free › Algebra › Polynomials › Multiply Polynomials

Multiply Polynomials

Multiply a polynomial by a monomial

Multiply Monomials

We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.

Example

Try it.

Multiply: ⓐ \((3{x}^{2})(-4{x}^{3})\) ⓑ \((\frac{5}{6}{x}^{3}y)(12x{y}^{2}).\)

Solution

ⓐ

\(\ (3{x}^{2})(-4{x}^{3})\)
Use the Commutative Property to rearrange the terms.\(\ 3\cdot (-4)\cdot {x}^{2}\cdot {x}^{3}\)
Multiply.\(\ -12{x}^{5}\)

ⓑ

\(\ (\frac{5}{6}{x}^{3}y)(12x{y}^{2})\)
Use the Commutative Property to rearrange the terms.\(\ \frac{5}{6}\cdot 12\cdot {x}^{3}\cdot x\cdot y\cdot {y}^{2}\)
Multiply.\(\ 10{x}^{4}{y}^{3}\)

Multiply a Polynomial by a Monomial

Multiplying a polynomial by a monomial is really just applying the Distributive Property.

Example

Try it.

Multiply: ⓐ \(-2y(4{y}^{2}+3y-5)\) ⓑ \(3{x}^{3}y({x}^{2}-8xy+{y}^{2}).\)

Solution

ⓐ

Distribute.\(\)
Multiply.

ⓑ

\(\ 3{x}^{3}y({x}^{2}-8xy+{y}^{2})\)
Distribute.\(\ 3{x}^{3}y\cdot {x}^{2}+(3{x}^{3}y)\cdot (-8xy)+(3{x}^{3}y)\cdot {y}^{2}\)
Multiply.\(\ 3{x}^{5}y-24{x}^{4}{y}^{2}+3{x}^{3}{y}^{3}\)

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

Example

Try it.

Multiply: ⓐ \((y+5)(y+8)\) ⓑ \((4y+3)(2y-5).\)

Solution

ⓐ

Distribute \((y+8).\)
Distribute again.
Combine like terms.

ⓑ

Distribute.
Distribute again.
Combine like terms.

If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms,

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.

Let’s multiply \((x+3)(x+7)\) using both methods.

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

Example

Try it.

Multiply: ⓐ \((y-7)(y+4)\) ⓑ \((4x+3)(2x-5).\)

Solution
  1. ⓐ

  2. ⓑ
Example

Try it.

Multiply: ⓐ \(({n}^{2}+4)(n-1)\) ⓑ \((3pq+5)(6pq-11).\)

Solution

ⓐ

Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms, there are none.

ⓑ

Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Multiply a Polynomial by a Polynomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.

Example

Try it.

Multiply \((b+3)(2{b}^{2}-5b+8)\) using ⓐ the Distributive Property and ⓑ the Vertical Method.

Solution

ⓐ

Distribute.
Multiply.
Combine like terms.

ⓑ It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Multiply \((2{b}^{2}-5b+8)\) by 3.
Multiply \((2{b}^{2}-5b+8)\) by \(b\).
Add like terms.

We have now seen two methods you can use to multiply a polynomial by a polynomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

Multiply Special Products

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for a pattern.

Look at these results. Do you see any patterns?

What about the number of terms? In each example we squared a binomial and the result was a trinomial.

\[{(a+b)}^{2}=\underline{\hspace{1.5em}}+\underline{\hspace{1.5em}}+\underline{\hspace{1.5em}}\]

Now look at the first term in each result. Where did it come from?

The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!

\[{(a+b)}^{2}={a}^{2}+\underline{\hspace{1.5em}}+\underline{\hspace{1.5em}}\]

To get the first term of the product, square the first term.

Where did the last term come from? Look at the examples and find the pattern.

\[{(a+b)}^{2}=\underline{\hspace{1.5em}}+\underline{\hspace{1.5em}}+{b}^{2}\]\[\begin{array}{l}{(a+b)}^{2}=\underline{\hspace{1.5em}}+2ab+\underline{\hspace{1.5em}} \\ {(a-b)}^{2}=\underline{\hspace{1.5em}}-2ab+\underline{\hspace{1.5em}}\end{array}\]
Example

Try it.

Multiply: ⓐ \({(x+5)}^{2}\) ⓑ \({(2x-3y)}^{2}.\)

Solution

ⓐ

Square the first term.
Square the last term.
Double their product.
Simplify.

ⓑ

Use the pattern.
Simplify.

\[(a+b)(a-b)={a}^{2}-\underline{\hspace{1.5em}}\]\[(a+b)(a-b)={a}^{2}-{b}^{2}\]
Example

Try it.

Multiply using the product of conjugates pattern: ⓐ \((2x+5)(2x-5)\) ⓑ \((5m-9n)(5m+9n).\)

Solution

ⓐ

Are the binomials conjugates?
It is the product of conjugates.
Square the first term, \(2x.\)
Square the last term, \(5.\)
Simplify. The product is a difference of squares.

ⓑ

This fits the pattern.
Use the pattern.
Simplify.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Multiply Polynomial Functions

Just as polynomials can be multiplied, polynomial functions can also be multiplied.

Example

Try it.

For functions \(f(x)=x+2\) and \(g(x)={x}^{2}-3x-4,\) find: ⓐ \((f\cdot g)(x)\) ⓑ \((f\cdot g)(2).\)

Solution

ⓐ

\((f\cdot g)(x)=f(x)\cdot g(x)\)
Substitute for \(f(x)\text{and}\ g(x).\)\((f\cdot g)(x)=(x+2)({x}^{2}-3x-4)\)
Multiply the polynomials.\((f\cdot g)(x)=x({x}^{2}-3x-4)+2({x}^{2}-3x-4)\)
Distribute.\((f\cdot g)(x)={x}^{3}-3{x}^{2}-4x+2{x}^{2}-6x-8\)
Combine like terms.\((f\cdot g)(x)={x}^{3}-{x}^{2}-10x-8\)

ⓑ In part ⓐ we found \((f\cdot g)(x)\) and now are asked to find \((f\cdot g)(2).\)

\((f\cdot g)(x)={x}^{3}-{x}^{2}-10x-8\)
To find \((f\cdot g)(2),\) substitute \(x=2.\)\((f\cdot g)(2)={2}^{3}-{2}^{2}-10\cdot 2-8\)
\((f\cdot g)(2)=8-4-20-8\)
\((f\cdot g)(2)=-24\)

Key Concepts

  • How to use the FOIL method to multiply two binomials.
  • Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
  • Multiplying a Polynomial by a Polynomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method
  • Binomial Squares Pattern
    If a and b are real numbers,
  • Product of Conjugates Pattern
    If \(a,b\) are real numbers

    The product is called a difference of squares.
    To multiply conjugates, square the first term, square the last term, write it as a difference of squares.
  • Comparing the Special Product Patterns
    Binomial SquaresProduct of Conjugates
    \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\)\((a-b)(a+b)={a}^{2}-{b}^{2}\)
    \({(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\)
    •  Squaring a binomial•  Multiplying conjugates
    •  Product is a trinomial•  Product is a binomial.
    •  Inner and outer terms with FOIL are the same.•  Inner and outer terms with FOIL are opposites.
    •  Middle term is double the product of the terms•  There is no middle term.
  • Multiplication of Polynomial Functions:
    • For functions \(f(x)\) and \(g(x),\)
      \[(f\cdot g)(x)=f(x)\cdot g(x)\]

Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

Try it.

ⓐ \((6{y}^{7})(-3{y}^{4})\)
ⓑ \((\frac{4}{7}r{s}^{2})(14r{s}^{3})\)

Try it.

ⓐ \((-10{x}^{5})(-3{x}^{3})\)
ⓑ \((\frac{5}{8}{x}^{3}y)(24{x}^{5}y)\)

Solution

ⓐ \(30{x}^{8}\) ⓑ \(15{x}^{8}{y}^{2}\)

Try it.

ⓐ \((-8{u}^{6})(-9u)\)
ⓑ \((\frac{2}{3}{x}^{2}y)(\frac{3}{4}x{y}^{2})\)

Try it.

ⓐ \((-6{c}^{4})(-12c)\)
ⓑ \((\frac{3}{5}{m}^{3}{n}^{2})(\frac{5}{9}{m}^{2}{n}^{3})\)

Solution

ⓐ \(72{c}^{5}\) ⓑ \(\frac{1}{3}{m}^{5}{n}^{5}\)

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Try it.

ⓐ \(-8x({x}^{2}+2x-15)\)
ⓑ \(5p{q}^{3}({p}^{2}-2pq+6{q}^{2})\)

Try it.

ⓐ \(-5t({t}^{2}+3t-18);\)
ⓑ \(9{r}^{3}s({r}^{2}-3rs+5{s}^{2})\)

Solution

ⓐ \(-5{t}^{3}-15{t}^{2}+90t\)
ⓑ \(9s{r}^{5}-27{s}^{2}{r}^{4}+45{s}^{3}{r}^{3}\)

Try it.

ⓐ \(-8y({y}^{2}+2y-15)\)
ⓑ \(-4{y}^{2}{z}^{2}(3{y}^{2}+12yz-{z}^{2})\)

Try it.

ⓐ \(-5m({m}^{2}+3m-18)\)
ⓑ \(-3{x}^{2}{y}^{2}(7{x}^{2}+10xy-{y}^{2})\)

Solution

ⓐ \(-5{m}^{3}-15{m}^{2}+90m\)
ⓑ \(-21{x}^{4}{y}^{2}-30{x}^{3}{y}^{3}+3{x}^{2}{y}^{4}\)

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.

Try it.

\((w+5)(w+7)\)

Try it.

\((y+9)(y+3)\)

Solution

\({y}^{2}+12y+27\)

Try it.

\((4p+11)(5p-4)\)

Try it.

\((7q+4)(3q-8)\)

Solution

\(21{q}^{2}-44q-32\)

In the following exercises, multiply the binomials. Use any method.

Try it.

\((x+8)(x+3)\)

Try it.

\((y-6)(y-2)\)

Solution

\({y}^{2}-8y+12\)

Try it.

\((2t-9)(10t+1)\)

Try it.

\((6p+5)(p+1)\)

Solution

\(6{p}^{2}+11p+5\)

Try it.

\((q-5)(q+8)\)

Try it.

\((m+11)(m-4)\)

Solution

\({m}^{2}+7m-44\)

Try it.

\((7m+1)(m-3)\)

Try it.

\((3r-8)(11r+1)\)

Solution

\(33{r}^{2}-85r-8\)

Try it.

\(({x}^{2}+3)(x+2)\)

Try it.

\(({y}^{2}-4)(y+3)\)

Solution

\({y}^{3}+3{y}^{2}-4y-12\)

Try it.

\((5ab-1)(2ab+3)\)

Try it.

\((2xy+3)(3xy+2)\)

Solution

\(6{x}^{2}{y}^{2}+13xy+6\)

Try it.

\(({x}^{2}+8)({x}^{2}-5)\)

Try it.

\(({y}^{2}-7)({y}^{2}-4)\)

Solution

\({y}^{4}-11{y}^{2}+28\)

Try it.

\((6pq-3)(4pq-5)\)

Try it.

\((3rs-7)(3rs-4)\)

Solution

\(9{r}^{2}{s}^{2}-33rs+28\)

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

Try it.

\((x+5)({x}^{2}+4x+3)\)

Try it.

\((u+4)({u}^{2}+3u+2)\)

Solution

\({u}^{3}+7{u}^{2}+14u+8\)

Try it.

\((y+8)(4{y}^{2}+y-7)\)

Try it.

\((a+10)(3{a}^{2}+a-5)\)

Solution

\(3{a}^{3}+31{a}^{2}+5a-50\)

Try it.

\(({y}^{2}-3y+8)(4{y}^{2}+y-7)\)

Try it.

\((2{a}^{2}-5a+10)(3{a}^{2}+a-5)\)

Solution

\(6{a}^{4}-13{a}^{3}+15{a}^{2}+35a-50\)

Multiply Special Products

In the following exercises, multiply. Use either method.

Try it.

\((w-7)({w}^{2}-9w+10)\)

Try it.

\((p-4)({p}^{2}-6p+9)\)

Solution

\({p}^{3}-10{p}^{2}+33p-36\)

Try it.

\((3q+1)({q}^{2}-4q-5)\)

Try it.

\((6r+1)({r}^{2}-7r-9)\)

Solution

\(6{r}^{3}-41{r}^{2}-61r-9\)

In the following exercises, square each binomial using the Binomial Squares Pattern.

Try it.

\({(w+4)}^{2}\)

Try it.

\({(q+12)}^{2}\)

Solution

\({q}^{2}+24q+144\)

Try it.

\({(3x-y)}^{2}\)

Try it.

\({(2y-3z)}^{2}\)

Solution

\(4{y}^{2}-12yz+9{z}^{2}\)

Try it.

\({(y+\frac{1}{4})}^{2}\)

Try it.

\({(x+\frac{2}{3})}^{2}\)

Solution

\({x}^{2}+\frac{4}{3}x+\frac{4}{9}\)

Try it.

\({(\frac{1}{5}x-\frac{1}{7}y)}^{2}\)

Try it.

\({(\frac{1}{8}x-\frac{1}{9}y)}^{2}\)

Solution

\(\frac{1}{64}{x}^{2}-\frac{1}{36}xy+\frac{1}{81}{y}^{2}\)

Try it.

\({(3{x}^{2}+2)}^{2}\)

Try it.

\({(5{u}^{2}+9)}^{2}\)

Solution

\(25{u}^{4}+90{u}^{2}+81\)

Try it.

\({(4{y}^{3}-2)}^{2}\)

Try it.

\({(8{p}^{3}-3)}^{2}\)

Solution

\(64{p}^{6}-48{p}^{3}+9\)

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

Try it.

\((5k+6)(5k-6)\)

Try it.

\((8j+4)(8j-4)\)

Solution

\(64{j}^{2}-16\)

Try it.

\((11k+4)(11k-4)\)

Try it.

\((9c+5)(9c-5)\)

Solution

\(81{c}^{2}-25\)

Try it.

\((9c-2d)(9c+2d)\)

Try it.

\((7w+10x)(7w-10x)\)

Solution

\(49{w}^{2}-100{x}^{2}\)

Try it.

\((m+\frac{2}{3}n)(m-\frac{2}{3}n)\)

Try it.

\((p+\frac{4}{5}q)(p-\frac{4}{5}q)\)

Solution

\({p}^{2}-\frac{16}{25}{q}^{2}\)

Try it.

\((ab-4)(ab+4)\)

Try it.

\((xy-9)(xy+9)\)

Solution

\({x}^{2}{y}^{2}-81\)

Try it.

\((12{p}^{3}-11{q}^{2})(12{p}^{3}+11{q}^{2})\)

Try it.

\((15{m}^{2}-8{n}^{4})(15{m}^{2}+8{n}^{4})\)

Solution

\(225{m}^{4}-64{n}^{8}\)

In the following exercises, find each product.

Try it.

\((p-3)(p+3)\)

Try it.

\({(t-9)}^{2}\)

Solution

\({t}^{2}-18t+81\)

Try it.

\({(m+n)}^{2}\)

Try it.

\((2x+y)(x-2y)\)

Solution

\(2{x}^{2}-3xy-2{y}^{2}\)

Try it.

\({(2r+12)}^{2}\)

Try it.

\((3p+8)(3p-8)\)

Solution

\(9{p}^{2}-64\)

Try it.

\((7a+b)(a-7b)\)

Try it.

\({(k-6)}^{2}\)

Solution

\({k}^{2}-12k+36\)

Try it.

\({({a}^{5}-7b)}^{2}\)

Try it.

\(({x}^{2}+8y)(8x-{y}^{2})\)

Solution

\(8{x}^{3}-{x}^{2}{y}^{2}+64xy-8{y}^{3}\)

Try it.

\(({r}^{6}+{s}^{6})({r}^{6}-{s}^{6})\)

Try it.

\({({y}^{4}+2z)}^{2}\)

Solution

\({y}^{8}+4{y}^{4}z+4{z}^{2}\)

Try it.

\(({x}^{5}+{y}^{5})({x}^{5}-{y}^{5})\)

Try it.

\({({m}^{3}-8n)}^{2}\)

Solution

\({m}^{6}-16{m}^{3}n+64{n}^{2}\)

Try it.

\({(9p+8q)}^{2}\)

Try it.

\(({r}^{2}-{s}^{3})({r}^{3}+{s}^{2})\)

Solution

\({r}^{5}+{r}^{2}{s}^{2}-{r}^{3}{s}^{3}-{s}^{5}\)

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like \(2(x-3)\). You multiplied both terms in the parentheses, \(x\ \text{and}\ 3\), by 2, to get \(2x-6\). With this chapter’s new vocabulary, you can say you were multiplying a binomial, \(x-3\), by a monomial, 2.

Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

Example

Try it.

Multiply: \(4(x+3).\)

Solution

Distribute.
Simplify.

Example

Try it.

Multiply: \(y(y-2).\)

Solution

Distribute.
Simplify.

Example

Try it.

Multiply: \(7x(2x+y).\)

Solution

Distribute.
Simplify.

Example

Try it.

Multiply: \(-2y(4{y}^{2}+3y-5).\)

Solution

Distribute.
Simplify.

Example

Try it.

Multiply: \(2{x}^{3}({x}^{2}-8x+1).\)

Solution

Distribute.
Simplify.

Example

Try it.

Multiply: \((x+3)p.\)

Solution

The monomial is the second factor.
Distribute.
Simplify.

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

Look at , where we multiplied a binomial by a monomial.

We distributed the p to get:
What if we have (x + 7) instead of p?
Distribute (x + 7).
Distribute again.
Combine like terms.

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial, four multiplications.

Example

Try it.

Multiply: \((y+5)(y+8).\)

Solution

Distribute (y + 8).
Distribute again
Combine like terms.

Example

Try it.

Multiply: \((2y+5)(3y+4).\)

Solution

Distribute (3y + 4).
Distribute again
Combine like terms.

Example

Try it.

Multiply: \((4y+3)(2y-5).\)

Solution

Distribute.
Distribute again.
Combine like terms.

Example

Try it.

Multiply: \((x-2)(x-y).\)

Solution

Distribute.
Distribute again.
There are no like terms to combine.

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

Example

Try it.

Multiply using the Distributive Property: \((b+3)(2{b}^{2}-5b+8).\)

Solution

Distribute.
Multiply.
Combine like terms.

Now let’s do this same multiplication using the Vertical Method.

Example

Try it.

Multiply using the Vertical Method: \((b+3)(2{b}^{2}-5b+8).\)

Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Multiply (2b2 − 5b + 8) by 3.
Multiply (2b2 − 5b + 8) by b.
Add like terms.

We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

Key Concepts

  • FOIL Method for Multiplying Two Binomials, To multiply two binomials:
    1. Multiply the First terms.
    2. Multiply the Outer terms.
    3. Multiply the Inner terms.
    4. Multiply the Last terms.
  • Multiplying Two Binomials, To multiply binomials, use the:
    • Distributive Property ()
    • FOIL Method ()
    • Vertical Method ()
  • Multiplying a Trinomial by a Binomial, To multiply a trinomial by a binomial, use the:
    • Distributive Property ()
    • Vertical Method ()

Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Try it.

\(\text{-}b(b+9)\)

Solution

\(\text{-}{b}^{2}-9b\)

Try it.

\(\text{-}y(y+3)\)

Try it.

\(\text{-}x(x-10)\)

Solution

\(\text{-}{x}^{2}+10x\)

Try it.

\(\text{-}p(p-15)\)

Try it.

\(6r(4r+s)\)

Solution

\(24{r}^{2}+6rs\)

Try it.

\(5c(9c+d)\)

Try it.

\(12x(x-10)\)

Solution

\(12{x}^{2}-120x\)

Try it.

\(9m(m-11)\)

Try it.

\(-9a(3a+5)\)

Solution

\(-27{a}^{2}-45a\)

Try it.

\(-4p(2p+7)\)

Try it.

\(3({p}^{2}+10p+25)\)

Solution

\(3{p}^{2}+30p+75\)

Try it.

\(6({y}^{2}+8y+16)\)

Try it.

\(-8x({x}^{2}+2x-15)\)

Solution

\(-8{x}^{3}-16{x}^{2}+120x\)

Try it.

\(-5t({t}^{2}+3t-18)\)

Try it.

\(5{q}^{3}({q}^{3}-2q+6)\)

Solution

\(5{q}^{6}-10{q}^{4}+30{q}^{3}\)

Try it.

\(4{x}^{3}({x}^{4}-3x+7)\)

Try it.

\(-8y({y}^{2}+2y-15)\)

Solution

\(-8{y}^{3}-16{y}^{2}+120y\)

Try it.

\(-5m({m}^{2}+3m-18)\)

Try it.

\(5{q}^{3}({q}^{2}-2q+6)\)

Solution

\(5{q}^{5}-10{q}^{4}+30{q}^{3}\)

Try it.

\(9{r}^{3}({r}^{2}-3r+5)\)

Try it.

\(-4{z}^{2}(3{z}^{2}+12z-1)\)

Solution

\(-12{z}^{4}-48{z}^{3}+4{z}^{2}\)

Try it.

\(-3{x}^{2}(7{x}^{2}+10x-1)\)

Try it.

\((w-6)\cdot 8\)

Solution

\(8w-48\)

Try it.

\((k-4)\cdot 5\)

Try it.

\(15(r-24)\)

Solution

\(15r-360\)

Try it.

\(12(v-30)\)

Try it.

\(-3(m+11)\)

Solution

\(-3m-33\)

Try it.

\(-4(p+15)\)

Try it.

\(n({n}^{2}-3n)\)

Solution

\({n}^{3}-3{n}^{2}\)

Try it.

\(s({s}^{2}-6s)\)

Try it.

\(6x(4x+y)\)

Solution

\(24{x}^{2}+6xy\)

Try it.

\(5a(9a+b)\)

Try it.

\(5p(11p-5q)\)

Solution

\(55{p}^{2}-25pq\)

Try it.

\(12u(3u-4v)\)

Try it.

\(3({v}^{2}+10v+25)\)

Solution

\(3{v}^{2}+30v+75\)

Try it.

\(6({x}^{2}+8x+16)\)

Try it.

\(2n(4{n}^{2}-4n+1)\)

Solution

\(8{n}^{3}-8{n}^{2}+2n\)

Try it.

\(3r(2{r}^{2}-6r+2)\)

Try it.

\(-8y({y}^{2}+2y-15)\)

Solution

\(-8{y}^{3}-16{y}^{2}+120y\)

Try it.

\(-5m({m}^{2}+3m-18)\)

Try it.

\(5{q}^{3}({q}^{2}-2q+6)\)

Solution

\(5{q}^{5}-10{q}^{4}+30{q}^{3}\)

Try it.

\(9{r}^{3}({r}^{2}-3r+5)\)

Try it.

\(-4{z}^{2}(3{z}^{2}+12z-1)\)

Solution

\(-12{z}^{4}-48{z}^{3}+4{z}^{2}\)

Try it.

\(-3{x}^{2}(7{x}^{2}+10x-1)\)

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

Try it.

\((w+5)(w+7)\)

Solution

\({w}^{2}+12w+35\)

Try it.

\((y+9)(y+3)\)

Try it.

\((p+11)(p-4)\)

Solution

\({p}^{2}+7p-44\)

Try it.

\((q+4)(q-8)\)

In the following exercises, multiply the binomials. Use any method.

Try it.

\((x+8)(x+3)\)

Solution

\({x}^{2}+11x+24\)

Try it.

\((y+7)(y+4)\)

Try it.

\((y-6)(y-2)\)

Solution

\({y}^{2}-8y+12\)

Try it.

\((x-7)(x-2)\)

Try it.

\((w-4)(w+7)\)

Solution

\({w}^{2}+3w-28\)

Try it.

\((q-5)(q+8)\)

Try it.

\((p+12)(p-5)\)

Solution

\({p}^{2}+7p-60\)

Try it.

\((m+11)(m-4)\)

Try it.

\((6p+5)(p+1)\)

Solution

\(6{p}^{2}+11p+5\)

Try it.

\((7m+1)(m+3)\)

Try it.

\((2t-9)(10t+1)\)

Solution

\(20{t}^{2}-88t-9\)

Try it.

\((3r-8)(11r+1)\)

Try it.

\((5x-y)(3x-6)\)

Solution

\(15{x}^{2}-3xy-30x+6y\)

Try it.

\((10a-b)(3a-4)\)

Try it.

\((a+b)(2a+3b)\)

Solution

\(2{a}^{2}+5ab+3{b}^{2}\)

Try it.

\((r+s)(3r+2s)\)

Try it.

\((4z-y)(z-6)\)

Solution

\(4{z}^{2}-24z-zy+6y\)

Try it.

\((5x-y)(x-4)\)

Try it.

\(({x}^{2}+3)(x+2)\)

Solution

\({x}^{3}+2{x}^{2}+3x+6\)

Try it.

\(({y}^{2}-4)(y+3)\)

Try it.

\(({x}^{2}+8)({x}^{2}-5)\)

Solution

\({x}^{4}+3{x}^{2}-40\)

Try it.

\(({y}^{2}-7)({y}^{2}-4)\)

Try it.

\((5ab-1)(2ab+3)\)

Solution

\(10{a}^{2}{b}^{2}+13ab-3\)

Try it.

\((2xy+3)(3xy+2)\)

Try it.

\((6pq-3)(4pq-5)\)

Solution

\(24{p}^{2}{q}^{2}-42pq+15\)

Try it.

\((3rs-7)(3rs-4)\)

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

Try it.

\((x+5)({x}^{2}+4x+3)\)

Solution

\({x}^{3}+9{x}^{2}+23x+15\)

Try it.

\((u+4)({u}^{2}+3u+2)\)

Try it.

\((y+8)(4{y}^{2}+y-7)\)

Solution

\(4{y}^{3}+33{y}^{2}+y-56\)

Try it.

\((a+10)(3{a}^{2}+a-5)\)

In the following exercises, multiply. Use either method.

Try it.

\((w-7)({w}^{2}-9w+10)\)

Solution

\({w}^{3}-16{w}^{2}+73w-70\)

Try it.

\((p-4)({p}^{2}-6p+9)\)

Try it.

\((3q+1)({q}^{2}-4q-5)\)

Solution

\(3{q}^{3}-11{q}^{2}-19q-5\)

Try it.

\((6r+1)({r}^{2}-7r-9)\)

Mixed Practice

Try it.

\((10y-6)+(4y-7)\)

Solution

\(14y-13\)

Try it.

\((15p-4)+(3p-5)\)

Try it.

\(({x}^{2}-4x-34)-({x}^{2}+7x-6)\)

Solution

\(-11x-28\)

Try it.

\(({j}^{2}-8j-27)-({j}^{2}+2j-12)\)

Try it.

\(5q(3{q}^{2}-6q+11)\)

Solution

\(15{q}^{3}-30{q}^{2}+55q\)

Try it.

\(8t(2{t}^{2}-5t+6)\)

Try it.

\((s-7)(s+9)\)

Solution

\({s}^{2}+2s-63\)

Try it.

\((x-5)(x+13)\)

Try it.

\(({y}^{2}-2y)(y+1)\)

Solution

\({y}^{3}-{y}^{2}-2y\)

Try it.

\(({a}^{2}-3a)(4a+5)\)

Try it.

\((3n-4)({n}^{2}+n-7)\)

Solution

\(3{n}^{3}-{n}^{2}-25n+28\)

Try it.

\((6k-1)({k}^{2}+2k-4)\)

Try it.

\((7p+10)(7p-10)\)

Solution

\(49{p}^{2}-100\)

Try it.

\((3y+8)(3y-8)\)

Try it.

\((4{m}^{2}-3m-7){m}^{2}\)

Solution

\(4{m}^{4}-3{m}^{3}-7{m}^{2}\)

Try it.

\((15{c}^{2}-4c+5){c}^{4}\)

Try it.

\((5a+7b)(5a+7b)\)

Solution

\(25{a}^{2}+70ab+49{b}^{2}\)

Try it.

\((3x-11y)(3x-11y)\)

Try it.

\((4y+12z)(4y-12z)\)

Solution

\(16{y}^{2}-144{z}^{2}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Multiply a Polynomial by a Monomial

In Distributive Property you learned to use the Distributive Property to simplify expressions such as \(2(x-3).\) You multiplied both terms in the parentheses, \(x\ \text{and}\ 3,\) by \(2,\) to get \(2x-6.\) With this chapter's new vocabulary, you can say you were multiplying a binomial, \(x-3,\) by a monomial, \(2.\) Multiplying a binomial by a monomial is nothing new for you!

Example

Try it.

Multiply: \(3(x+7).\)

Solution
\(3(x+7)\)
Distribute.
\(3\cdot x+3\cdot 7\)
Simplify.\(3x+21\)
Example

Try it.

Multiply: \(x(x-8).\)

Solution
Distribute.
Simplify.
Example

Try it.

Multiply: \(10x(4x+y).\)

Solution
Distribute.
Simplify.

Multiplying a monomial by a trinomial works in much the same way.

Example

Try it.

Multiply: \(-2x(5{x}^{2}+7x-3).\)

Solution
\(-2x(5{x}^{2}+7x-3)\)
Distribute.
\(-2x⋅5{x}^{2}+(-2x)⋅7x-(-2x)⋅3\)
Simplify.\(-10{x}^{3}-14{x}^{2}+6x\)
Example

Try it.

Multiply: \(4{y}^{3}({y}^{2}-8y+1).\)

Solution
\(4{y}^{3}({y}^{2}-8y+1)\)
Distribute.
\(4{y}^{3}⋅{y}^{2}-4{y}^{3}⋅8y+4{y}^{3}⋅1\)
Simplify.\(4{y}^{5}-32{y}^{4}+4{y}^{3}\)

Now we will have the monomial as the second factor.

Example

Try it.

Multiply: \((x+3)p.\)

Solution
\((x+3)p\)
Distribute.
\(x⋅p+3⋅p\)
Simplify.\(xp+3p\)

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

We will start by using the Distributive Property. Look again at .

We distributed the \(p\) to get
What if we have \((x+7)\) instead of \(p\)?
Distribute \((x+7)\).
Distribute again.\({x}^{2}+7x+3x+21\)
Combine like terms.\({x}^{2}+10x+21\)

Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial, four multiplications.

Be careful to distinguish between a sum and a product.

\[\begin{array}{llll}\text{Sum} & & & \text{Product} \\ x+x & & & x\cdot x \\ 2x & & & {x}^{2} \\ \text{combine like terms} & & & \text{add exponents of like bases}\end{array}\]
Example

Try it.

Multiply: \((x+6)(x+8).\)

Solution
\((x+6)(x+8)\)
Distribute \((x+8)\).
Distribute again.\({x}^{2}+8x+6x+48\)
Simplify.\({x}^{2}+14x+48\)

Now we'll see how to multiply binomials where the variable has a coefficient.

Example

Try it.

Multiply: \((2x+9)(3x+4).\)

Solution
\((2x+9)(3x+4)\)
Distribute. \((3x+4)\)
Distribute again.\(6{x}^{2}+8x+27x+36\)
Simplify.\(6{x}^{2}+35x+36\)

In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

Example

Try it.

Multiply: \((4y+3)(6y-5).\)

Solution
\((4y+3)(6y-5)\)
Distribute.
Distribute again.\(24{y}^{2}-20y+18y-15\)
Simplify.\(24{y}^{2}-2y-15\)

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

Example

Try it.

Multiply: \((x+2)(x-y).\)

Solution
Distribute.
Distribute again.
Simplify.There are no like terms to combine.

Condensed: the full section is in OpenStax Prealgebra 2e.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

Example

Try it.

Multiply using the Distributive Property: \((x+3)(2{x}^{2}-5x+8).\)

Solution
Distribute.
Multiply.\(2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24\)
Combine like terms.\(2{x}^{3}+{x}^{2}-7x+24\)

Now let's do this same multiplication using the Vertical Method.

Example

Try it.

Multiply using the Vertical Method: \((x+3)(2{x}^{2}-5x+8).\)

Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Multiply \((2{x}^{2}-5x+8)\) by 3.
Multiply \((2{x}^{2}-5x+8)\) by \(x\).
Add like terms.

Key Concepts

  • Use the FOIL method for multiplying two binomials.
    Step 1. Multiply the First terms.
    Step 2. Multiply the Outer terms.
    Step 3. Multiply the Inner terms.
    Step 4. Multiply the Last terms.
    Step 5. Combine like terms, when possible.
  • Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
    • Vertical Method
  • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method

Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Try it.

\(15(r-24)\)

Solution

15r − 360

Try it.

\(12(v-30)\)

Try it.

\(-3(m+11)\)

Solution

−3m − 33

Try it.

\(-4(p+15)\)

Try it.

\(n({n}^{2}-3n)\)

Solution

n3 − 3n2

Try it.

\(s({s}^{2}-6s)\)

Try it.

\(12x(x-10)\)

Solution

12x2 − 120x

Try it.

\(9m(m-11)\)

Try it.

\(-9a(3a+5)\)

Solution

−27a2 − 45a

Try it.

\(-4p(2p+7)\)

Try it.

\(6x(4x+y)\)

Solution

24x2 + 6xy

Try it.

\(5a(9a+b)\)

Try it.

\(5p(11p-5q)\)

Solution

55p2 − 25pq

Try it.

\(12u(3u-4v)\)

Try it.

\(3({v}^{2}+10v+25)\)

Solution

3v2 + 30v + 75

Try it.

\(6({x}^{2}+8x+16)\)

Try it.

\(2n(4{n}^{2}-4n+1)\)

Solution

8n3 − 8n2 + 2n

Try it.

\(3r(2{r}^{2}-6r+2)\)

Try it.

\(-8y({y}^{2}+2y-15)\)

Solution

−8y3 − 16y2 + 120y

Try it.

\(-5m({m}^{2}+3m-18)\)

Try it.

\(5{q}^{3}({q}^{2}-2q+6)\)

Solution

5q5 − 10q4 + 30q3

Try it.

\(9{r}^{3}({r}^{2}-3r+5)\)

Try it.

\(-4{z}^{2}(3{z}^{2}+12z-1)\)

Solution

−12z4 − 48z3 + 4z2

Try it.

\(-3{x}^{2}(7{x}^{2}+10x-1)\)

Try it.

\((w-6)\ \cdot \ 8\)

Solution

8w − 48

Try it.

\((k-4)\ \cdot \ 5\)

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method

Try it.

\((x+4)(x+6)\)

Solution

x2 + 10x + 24

Try it.

\((u+8)(u+2)\)

Try it.

\((n+12)(n-3)\)

Solution

n2 + 9n − 36

Try it.

\((y+3)(y-9)\)

In the following exercises, multiply the following binomials. Use any method.

Try it.

\((y+8)(y+3)\)

Solution

y2 + 11y + 24

Try it.

\((x+5)(x+9)\)

Try it.

\((a+6)(a+16)\)

Solution

a2 + 22a + 96

Try it.

\((q+8)(q+12)\)

Try it.

\((u-5)(u-9)\)

Solution

u2 − 14u + 45

Try it.

\((r-6)(r-2)\)

Try it.

\((z-10)(z-22)\)

Solution

z2 − 32z + 220

Try it.

\((b-5)(b-24)\)

Try it.

\((x-4)(x+7)\)

Solution

x2 + 3x − 28

Try it.

\((s-3)(s+8)\)

Try it.

\((v+12)(v-5)\)

Solution

v2 + 7v − 60

Try it.

\((d+15)(d-4)\)

Try it.

\((6n+5)(n+1)\)

Solution

6n2 + 11n + 5

Try it.

\((7y+1)(y+3)\)

Try it.

\((2m-9)(10m+1)\)

Solution

20m2 − 88m − 9

Try it.

\((5r-4)(12r+1)\)

Try it.

\((4c-1)(4c+1)\)

Solution

16c2 − 1

Try it.

\((8n-1)(8n+1)\)

Try it.

\((3u-8)(5u-14)\)

Solution

15u2 − 82u + 112

Try it.

\((2q-5)(7q-11)\)

Try it.

\((a+b)(2a+3b)\)

Solution

2a2 + 5ab + 3b2

Try it.

\((r+s)(3r+2s)\)

Try it.

\((5x-y)(x-4)\)

Solution

5x2 − 20x − xy + 4y

Try it.

\((4z-y)(z-6)\)

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method.

Try it.

\((u+4)({u}^{2}+3u+2)\)

Solution

u3 + 7u2 + 14u + 8

Try it.

\((x+5)({x}^{2}+8x+3)\)

Try it.

\((a+10)(3{a}^{2}+a-5)\)

Solution

3a3 + 31a2 + 5a − 50

Try it.

\((n+8)(4{n}^{2}+n-7)\)

In the following exercises, multiply. Use either method.

Try it.

\((y-6)({y}^{2}-10y+9)\)

Solution

y3 − 16y2 + 69y − 54

Try it.

\((k-3)({k}^{2}-8k+7)\)

Try it.

\((2x+1)({x}^{2}-5x-6)\)

Solution

2x3 − 9x2 − 17x − 6

Try it.

\((5v+1)({v}^{2}-6v-10)\)

Condensed: the full section is in OpenStax Prealgebra 2e.

Ngayon ikaw Walang kalkulador settles ito isa, ngunit ang mga piraso ng mga ito ay computable. Subukan ang isa sa ibaba, o type ang iyong sarili.

Pagsasanay (40)

Subukan ang bawat isa sa papel unang. Ipahayag ang sagot upang suriin; na-verify na mga maaaring buksan sa solver para sa bawat hakbang.

  1. Distribute: \(2(x+3).\)

    Ipahayag ang sagot

    \(2x+6\)

  2. Simplify: ⓐ \({9}^{2}\) ⓑ \({(-9)}^{2}\) ⓒ \(\text{-}{9}^{2}.\)

    Ipahayag ang sagot

    ⓐ \(81\); ⓑ \(81\); ⓒ \(-81\)

  3. Evaluate: \(2{x}^{2}-5x+3\) for \(x=-2.\)

    Ipahayag ang sagot

    \(21\)

  4. Multiply: ⓐ \((3{x}^{2})(-4{x}^{3})\) ⓑ \((\frac{5}{6}{x}^{3}y)(12x{y}^{2}).\)

    Ipahayag ang sagot

    ⓐ

    \(\ (3{x}^{2})(-4{x}^{3})\)
    Use the Commutative Property to rearrange the terms.\(\ 3\cdot (-4)\cdot {x}^{2}\cdot {x}^{3}\)
    Multiply.\(\ -12{x}^{5}\)

    ⓑ

    \(\ (\frac{5}{6}{x}^{3}y)(12x{y}^{2})\)
    Use the Commutative Property to rearrange the terms.\(\ \frac{5}{6}\cdot 12\cdot {x}^{3}\cdot x\cdot y\cdot {y}^{2}\)
    Multiply.\(\ 10{x}^{4}{y}^{3}\)

  5. Multiply: ⓐ \((5{y}^{7})(-7{y}^{4})\) ⓑ \((\frac{2}{5}{a}^{4}{b}^{3})(15a{b}^{3}).\)

    Ipahayag ang sagot

    ⓐ \(-35{y}^{11}\) ⓑ \(6{a}^{5}{b}^{6}\)

  6. Multiply: ⓐ \((-6{b}^{4})(-9{b}^{5})\) ⓑ \((\frac{2}{3}{r}^{5}s)(12{r}^{6}{s}^{7}).\)

    Ipahayag ang sagot

    ⓐ \(54{b}^{9}\) ⓑ \(8{r}^{11}{s}^{8}\)

  7. Multiply: ⓐ \(-2y(4{y}^{2}+3y-5)\) ⓑ \(3{x}^{3}y({x}^{2}-8xy+{y}^{2}).\)

    Ipahayag ang sagot

    ⓐ

    Distribute.\(\)
    Multiply.

    ⓑ

    \(\ 3{x}^{3}y({x}^{2}-8xy+{y}^{2})\)
    Distribute.\(\ 3{x}^{3}y\cdot {x}^{2}+(3{x}^{3}y)\cdot (-8xy)+(3{x}^{3}y)\cdot {y}^{2}\)
    Multiply.\(\ 3{x}^{5}y-24{x}^{4}{y}^{2}+3{x}^{3}{y}^{3}\)

  8. Multiply: ⓐ \(-3y(5{y}^{2}+8y-7)\) ⓑ \(4{x}^{2}{y}^{2}(3{x}^{2}-5xy+3{y}^{2}).\)

    Ipahayag ang sagot

    ⓐ \(-15{y}^{3}-24{y}^{2}+21y\)
    ⓑ \(12{x}^{4}{y}^{2}-20{x}^{3}{y}^{3}+12{x}^{2}{y}^{4}\)

  9. Multiply: ⓐ \(4{x}^{2}(2{x}^{2}-3x+5)\) ⓑ \(-6{a}^{3}b(3{a}^{2}-2ab+6{b}^{2}).\)

    Ipahayag ang sagot

    ⓐ \(8{x}^{4}-12{x}^{3}+20{x}^{2}\)
    ⓑ \(-18{a}^{5}b+12{a}^{4}{b}^{2}-36{a}^{3}{b}^{3}\)

  10. Multiply: ⓐ \((y+5)(y+8)\) ⓑ \((4y+3)(2y-5).\)

    Ipahayag ang sagot

    ⓐ

    Distribute \((y+8).\)
    Distribute again.
    Combine like terms.

    ⓑ

    Distribute.
    Distribute again.
    Combine like terms.

  11. Multiply: ⓐ \((x+8)(x+9)\) ⓑ \((3c+4)(5c-2).\)

    Ipahayag ang sagot

    ⓐ \({x}^{2}+17x+72\)
    ⓑ \(15{c}^{2}+14c-8\)

  12. Multiply: ⓐ \((5x+9)(4x+3)\) ⓑ \((5y+2)(6y-3).\)

    Ipahayag ang sagot

    ⓐ \(20{x}^{2}+51x+27\)
    ⓑ \(30{y}^{2}-3y-6\)

  13. Multiply: ⓐ \((y-7)(y+4)\) ⓑ \((4x+3)(2x-5).\)

    Ipahayag ang sagot
    1. ⓐ

    2. ⓑ

  14. Multiply: ⓐ \((x-7)(x+5)\) ⓑ \((3x+7)(5x-2).\)

    Ipahayag ang sagot

    ⓐ \({x}^{2}-2x-35\)
    ⓑ \(15{x}^{2}+29x-14\)

  15. Multiply: ⓐ \((b-3)(b+6)\) ⓑ \((4y+5)(4y-10).\)

    Ipahayag ang sagot

    ⓐ \({b}^{2}+3b-18\)
    ⓑ \(16{y}^{2}-20y-50\)

  16. Multiply: ⓐ \(({n}^{2}+4)(n-1)\) ⓑ \((3pq+5)(6pq-11).\)

    Ipahayag ang sagot

    ⓐ

    Step 1. Multiply the First terms.
    Step 2. Multiply the Outer terms.
    Step 3. Multiply the Inner terms.
    Step 4. Multiply the Last terms.
    Step 5. Combine like terms, there are none.

    ⓑ

    Step 1. Multiply the First terms.
    Step 2. Multiply the Outer terms.
    Step 3. Multiply the Inner terms.
    Step 4. Multiply the Last terms.
    Step 5. Combine like terms.

  17. Multiply: ⓐ \(({x}^{2}+6)(x-8)\) ⓑ \((2ab+5)(4ab-4).\)

    Ipahayag ang sagot

    ⓐ \({x}^{3}-8{x}^{2}+6x-48\)
    ⓑ \(8{a}^{2}{b}^{2}+12ab-20\)

  18. Multiply: ⓐ \(({y}^{2}+7)(y-9)\) ⓑ \((2xy+3)(4xy-5).\)

    Ipahayag ang sagot

    ⓐ \({y}^{3}-9{y}^{2}+7y-63\)
    ⓑ \(8{x}^{2}{y}^{2}+2xy-15\)

  19. Multiply using the Vertical Method: \((3y-1)(2y-6).\)

    Ipahayag ang sagot

    It does not matter which binomial goes on the top.


    Multiply \(3y-1\ \text{by}\ -6.\)
    Multiply \(3y-1\ \text{by}\ 2y.\)
    Add like terms.    		\(\begin{array}{l}3y-1 \\ \underline{\ \times \ 2y-6} \\ -18y+6 \\ \underline{6{y}^{2}-\ 2y\ } \\ 6{y}^{2}-20y+6\end{array}\)
    partial product
    partial product
    product

    Notice the partial products are the same as the terms in the FOIL method.

  20. Multiply using the Vertical Method: \((5m-7)(3m-6).\)

    Ipahayag ang sagot

    \(15{m}^{2}-51m+42\)

  21. Multiply using the Vertical Method: \((6b-5)(7b-3).\)

    Ipahayag ang sagot

    \(42{b}^{2}-53b+15\)

  22. Multiply \((b+3)(2{b}^{2}-5b+8)\) using ⓐ the Distributive Property and ⓑ the Vertical Method.

    Ipahayag ang sagot

    ⓐ

    Distribute.
    Multiply.
    Combine like terms.

    ⓑ It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

    Multiply \((2{b}^{2}-5b+8)\) by 3.
    Multiply \((2{b}^{2}-5b+8)\) by \(b\).
    Add like terms.

  23. Multiply\((y-3)({y}^{2}-5y+2)\) using ⓐ the Distributive Property and ⓑ the Vertical Method.

    Ipahayag ang sagot

    ⓐ \({y}^{3}-8{y}^{2}+17y-6\)
    ⓑ \({y}^{3}-8{y}^{2}+17y-6\)

  24. Multiply \((x+4)(2{x}^{2}-3x+5)\) using ⓐ the Distributive Property and ⓑ The Vertical Method.

    Ipahayag ang sagot

    ⓐ \(2{x}^{3}+5{x}^{2}-7x+20\)
    ⓑ \({y}^{3}-8{y}^{2}+17y-6\)

  25. Multiply: ⓐ \({(x+5)}^{2}\) ⓑ \({(2x-3y)}^{2}.\)

    Ipahayag ang sagot

    ⓐ

    Square the first term.
    Square the last term.
    Double their product.
    Simplify.

    ⓑ

    Use the pattern.
    Simplify.

  26. Multiply: ⓐ \({(x+9)}^{2}\) ⓑ \({(2c-d)}^{2}.\)

    Ipahayag ang sagot

    ⓐ \({x}^{2}+18x+81\)
    ⓑ \(4{c}^{2}-4cd+{d}^{2}\)

  27. Multiply: ⓐ \({(y+11)}^{2}\) ⓑ \({(4x-5y)}^{2}.\)

    Ipahayag ang sagot

    ⓐ \({y}^{2}+22y+121\)
    ⓑ \(16{x}^{2}-40xy+25{y}^{2}\)

  28. Multiply using the product of conjugates pattern: ⓐ \((2x+5)(2x-5)\) ⓑ \((5m-9n)(5m+9n).\)

    Ipahayag ang sagot

    ⓐ

    Are the binomials conjugates?
    It is the product of conjugates.
    Square the first term, \(2x.\)
    Square the last term, \(5.\)
    Simplify. The product is a difference of squares.

    ⓑ

    This fits the pattern.
    Use the pattern.
    Simplify.

  29. Multiply: ⓐ \((6x+5)(6x-5)\) ⓑ \((4p-7q)(4p+7q).\)

    Ipahayag ang sagot

    ⓐ \(36{x}^{2}-25\)
    ⓑ \(16{p}^{2}-49{q}^{2}\)

  30. Multiply: ⓐ \((2x+7)(2x-7)\) ⓑ \((3x-y)(3x+y).\)

    Ipahayag ang sagot

    ⓐ \(4{x}^{2}-49\) ⓑ \(9{x}^{2}-{y}^{2}\)

  31. Choose the appropriate pattern and use it to find the product:

    ⓐ \((2x-3)(2x+3)\) ⓑ \({(8x-5)}^{2}\) ⓒ \({(6m+7)}^{2}\) ⓓ \((5x-6)(6x+5).\)

    Ipahayag ang sagot

    ⓐ \((2x-3)(2x+3)\)

    These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.

    Use the pattern.
    Simplify.

    ⓑ \({(8x-5)}^{2}\)

    We are asked to square a binomial. It fits the binomial squares pattern.

    Use the pattern.
    Simplify.

    ⓒ \({(6m+7)}^{2}\)

    Again, we will square a binomial so we use the binomial squares pattern.

    Use the pattern.
    Simplify.

    ⓓ \((5x-6)(6x+5)\)

    This product does not fit the patterns, so we will use FOIL.

    \(\begin{array}{llll} & & & \ (5x-6)(6x+5) \\ \text{Use FOIL.} & & & \ 30{x}^{2}+25x-36x-30 \\ \text{Simplify.} & & & \ 30{x}^{2}-11x-30\end{array}\)

  32. Choose the appropriate pattern and use it to find the product:

    ⓐ \((9b-2)(2b+9)\) ⓑ \({(9p-4)}^{2}\) ⓒ \({(7y+1)}^{2}\) ⓓ \((4r-3)(4r+3).\)

    Ipahayag ang sagot

    ⓐ FOIL; \(18{b}^{2}+77b-18\)
    ⓑ Binomial Squares; \(81{p}^{2}-72p+16\)
    ⓒ Binomial Squares; \(49{y}^{2}+14y+1\)
    ⓓ Product of Conjugates; \(16{r}^{2}-9\)

  33. Choose the appropriate pattern and use it to find the product:

    ⓐ \({(6x+7)}^{2}\) ⓑ \((3x-4)(3x+4)\) ⓒ \((2x-5)(5x-2)\) ⓓ \({(6n-1)}^{2}.\)

    Ipahayag ang sagot

    ⓐ Binomial Squares; \(36{x}^{2}+84x+49\) ⓑ Product of Conjugates; \(9{x}^{2}-16\) ⓒ FOIL; \(10{x}^{2}-29x+10\) ⓓ Binomial Squares; \(36{n}^{2}-12n+1\)

  34. For functions \(f(x)=x+2\) and \(g(x)={x}^{2}-3x-4,\) find: ⓐ \((f\cdot g)(x)\) ⓑ \((f\cdot g)(2).\)

    Ipahayag ang sagot

    ⓐ

    \((f\cdot g)(x)=f(x)\cdot g(x)\)
    Substitute for \(f(x)\text{and}\ g(x).\)\((f\cdot g)(x)=(x+2)({x}^{2}-3x-4)\)
    Multiply the polynomials.\((f\cdot g)(x)=x({x}^{2}-3x-4)+2({x}^{2}-3x-4)\)
    Distribute.\((f\cdot g)(x)={x}^{3}-3{x}^{2}-4x+2{x}^{2}-6x-8\)
    Combine like terms.\((f\cdot g)(x)={x}^{3}-{x}^{2}-10x-8\)

    ⓑ In part ⓐ we found \((f\cdot g)(x)\) and now are asked to find \((f\cdot g)(2).\)

    \((f\cdot g)(x)={x}^{3}-{x}^{2}-10x-8\)
    To find \((f\cdot g)(2),\) substitute \(x=2.\)\((f\cdot g)(2)={2}^{3}-{2}^{2}-10\cdot 2-8\)
    \((f\cdot g)(2)=8-4-20-8\)
    \((f\cdot g)(2)=-24\)

  35. For functions \(f(x)=x-5\) and \(g(x)={x}^{2}-2x+3,\) find ⓐ \((f\cdot g)(x)\) ⓑ \((f\cdot g)(2).\)

    Ipahayag ang sagot

    ⓐ \((f\cdot g)(x)={x}^{3}-7{x}^{2}+13x-15\)
    ⓑ \((f\cdot g)(2)=-9\)

  36. For functions \(f(x)=x-7\) and \(g(x)={x}^{2}+8x+4,\) find ⓐ \((f\cdot g)(x)\) ⓑ \((f\cdot g)(2).\)

    Ipahayag ang sagot

    ⓐ \((f\cdot g)(x)={x}^{3}+{x}^{2}-52x-28\)
    ⓑ \((f\cdot g)(2)=-120\)

  37. ⓐ \((6{y}^{7})(-3{y}^{4})\)
    ⓑ \((\frac{4}{7}r{s}^{2})(14r{s}^{3})\)

  38. ⓐ \((-10{x}^{5})(-3{x}^{3})\)
    ⓑ \((\frac{5}{8}{x}^{3}y)(24{x}^{5}y)\)

    Ipahayag ang sagot

    ⓐ \(30{x}^{8}\) ⓑ \(15{x}^{8}{y}^{2}\)

  39. ⓐ \((-8{u}^{6})(-9u)\)
    ⓑ \((\frac{2}{3}{x}^{2}y)(\frac{3}{4}x{y}^{2})\)

  40. ⓐ \((-6{c}^{4})(-12c)\)
    ⓑ \((\frac{3}{5}{m}^{3}{n}^{2})(\frac{5}{9}{m}^{2}{n}^{3})\)

    Ipahayag ang sagot

    ⓐ \(72{c}^{5}\) ⓑ \(\frac{1}{3}{m}^{5}{n}^{5}\)

Panatilihin ang iyong sariling nagtatrabaho

Ang isang libreng account ay nagdaragdag ng mga tala sa bawat aralin, isang tala ng kung ano ang iyong natapos, ang iyong mga nalulutas na problema sa isang lugar, at isang guro na maaari mong tanungin tungkol sa pahinang ito. Ang matematika mismo ay bukas sa lahat, naka-sign in o hindi.

Mag-sign Up Mag-login

Mga simbolong ginagamit dito

I-tap ang anumang simbolo para sa buong kahulugan, isang larawan, at kung ano ang ibig sabihin ng bawat titik sa ito.

Paano: Multiply Polynomials

  1. Multiply a polynomial by a monomial
  2. Multiply a binomial by a binomial
  3. Multiply a trinomial by a binomial
  4. Multiply the
  5. Multiply the
  6. Multiply the
  7. Multiply the
  8. Combine like terms, when possible.

Mga katanungan na tanong ng mga tao

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Bahagi ng pahinang ito ay naaayon mula sa OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed at muling ipinaliwanag dito; errors ay aming. Dahil ang pinagmulan nito ay lisensyado para sa hindi pang-komersyal na paggamit, ang aralin na ito ay libre, hindi nagbebenta ng anuman, at ibinabahagi sa ilalim ng parehong lisensya.

Higit pa sa Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value