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Multiply Polynomials

Multiply monomials

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

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.

\[{(a+b)}^{2}=___+___+___\]

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!

\[{(a+b)}^{2}={a}^{2}+___+___\]

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.

\[{(a+b)}^{2}=___+___+{b}^{2}\]\[\begin{array}{l}{(a+b)}^{2}=___+2ab+___ \\ {(a-b)}^{2}=___-2ab+___\end{array}\]

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.

\[(a+b)(a-b)={a}^{2}-___\]\[(a+b)(a-b)={a}^{2}-{b}^{2}\]

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 Squares	Product 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.

\(4(w+10)\)

Solution

\(4w+40\)

Try it.

\(6(b+8)\)

Try it.

\(-3(a+7)\)

Solution

\(-3a-21\)

Try it.

\(-5(p+9)\)

Try it.

\(2(x-7)\)

Solution

\(2x-14\)

Try it.

\(7(y-4)\)

Try it.

\(-3(k-4)\)

Solution

\(-3k+12\)

Try it.

\(-8(j-5)\)

Try it.

\(q(q+5)\)

Solution

\({q}^{2}+5q\)

Try it.

\(k(k+7)\)

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.

\((2m-9)m\)

Solution

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

Try it.

\((8j-1)j\)

Try it.

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

Solution

\(8w-48\)

Try it.

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

Try it.

\(4(x+10)\)

Solution

\(4x+40\)

Try it.

\(6(y+8)\)

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.

\(-8(z-5)\)

Solution

\(-8z+40\)

Try it.

\(-3(x-9)\)

Try it.

\(u(u+5)\)

Solution

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

Try it.

\(q(q+7)\)

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.

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

Try it.

\((2y-9)y\)

Solution

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

Try it.

\((8b-1)b\)

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.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Distribute: \(2(x+3).\)
If you missed this problem, review .

Révèle la réponse

\(2x+6\)
Simplify: ⓐ \({9}^{2}\) ⓑ \({(-9)}^{2}\) ⓒ \(\text{-}{9}^{2}.\)
If you missed this problem, review .

Révèle la réponse

ⓐ \(81\); ⓑ \(81\); ⓒ \(-81\)
Evaluate: \(2{x}^{2}-5x+3\) for \(x=-2.\)
If you missed this problem, review .

Révèle la réponse

\(21\)
Multiply: ⓐ \((3{x}^{2})(-4{x}^{3})\) ⓑ \((\frac{5}{6}{x}^{3}y)(12x{y}^{2}).\)

Révèle la réponse

ⓐ

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

Révèle la réponse

ⓐ \(-35{y}^{11}\) ⓑ \(6{a}^{5}{b}^{6}\)
Multiply: ⓐ \((-6{b}^{4})(-9{b}^{5})\) ⓑ \((\frac{2}{3}{r}^{5}s)(12{r}^{6}{s}^{7}).\)

Révèle la réponse

ⓐ \(54{b}^{9}\) ⓑ \(8{r}^{11}{s}^{8}\)
Multiply: ⓐ \(-2y(4{y}^{2}+3y-5)\) ⓑ \(3{x}^{3}y({x}^{2}-8xy+{y}^{2}).\)

Révèle la réponse

ⓐ

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

Révèle la réponse

ⓐ \(-15{y}^{3}-24{y}^{2}+21y\)
ⓑ \(12{x}^{4}{y}^{2}-20{x}^{3}{y}^{3}+12{x}^{2}{y}^{4}\)
Multiply: ⓐ \(4{x}^{2}(2{x}^{2}-3x+5)\) ⓑ \(-6{a}^{3}b(3{a}^{2}-2ab+6{b}^{2}).\)

Révèle la réponse

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

Révèle la réponse

ⓐ

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

Distribute.
Distribute again.
Combine like terms.
Multiply: ⓐ \((x+8)(x+9)\) ⓑ \((3c+4)(5c-2).\)

Révèle la réponse

ⓐ \({x}^{2}+17x+72\)
ⓑ \(15{c}^{2}+14c-8\)
Multiply: ⓐ \((5x+9)(4x+3)\) ⓑ \((5y+2)(6y-3).\)

Révèle la réponse

ⓐ \(20{x}^{2}+51x+27\)
ⓑ \(30{y}^{2}-3y-6\)
Multiply: ⓐ \((y-7)(y+4)\) ⓑ \((4x+3)(2x-5).\)
Révèle la réponse
1. ⓐ
2. ⓑ
Multiply: ⓐ \((x-7)(x+5)\) ⓑ \((3x+7)(5x-2).\)

Révèle la réponse

ⓐ \({x}^{2}-2x-35\)
ⓑ \(15{x}^{2}+29x-14\)
Multiply: ⓐ \((b-3)(b+6)\) ⓑ \((4y+5)(4y-10).\)

Révèle la réponse

ⓐ \({b}^{2}+3b-18\)
ⓑ \(16{y}^{2}-20y-50\)
Multiply: ⓐ \(({n}^{2}+4)(n-1)\) ⓑ \((3pq+5)(6pq-11).\)

Révèle la réponse

ⓐ

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

Révèle la réponse

ⓐ \({x}^{3}-8{x}^{2}+6x-48\)
ⓑ \(8{a}^{2}{b}^{2}+12ab-20\)
Multiply: ⓐ \(({y}^{2}+7)(y-9)\) ⓑ \((2xy+3)(4xy-5).\)

Révèle la réponse

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

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

Révèle la réponse

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.

\(\begin{array}{l}3y-1 \\ \underset{___________}{\ \times \ 2y-6} \\ -18y+6 \\ \underset{___________}{6{y}^{2}-\ 2y\ } \\ 6{y}^{2}-20y+6\end{array}\)

partial product
partial product
product

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

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

Révèle la réponse

\(15{m}^{2}-51m+42\)
Multiply using the Vertical Method: \((6b-5)(7b-3).\)

Révèle la réponse

\(42{b}^{2}-53b+15\)
Multiply \((b+3)(2{b}^{2}-5b+8)\) using ⓐ the Distributive Property and ⓑ the Vertical Method.

Révèle la réponse

ⓐ

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

Révèle la réponse

ⓐ \({y}^{3}-8{y}^{2}+17y-6\)
ⓑ \({y}^{3}-8{y}^{2}+17y-6\)
Multiply \((x+4)(2{x}^{2}-3x+5)\) using ⓐ the Distributive Property and ⓑ The Vertical Method.

Révèle la réponse

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

Révèle la réponse

ⓐ

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

Use the pattern.
Simplify.
Multiply: ⓐ \({(x+9)}^{2}\) ⓑ \({(2c-d)}^{2}.\)

Révèle la réponse

ⓐ \({x}^{2}+18x+81\)
ⓑ \(4{c}^{2}-4cd+{d}^{2}\)
Multiply: ⓐ \({(y+11)}^{2}\) ⓑ \({(4x-5y)}^{2}.\)

Révèle la réponse

ⓐ \({y}^{2}+22y+121\)
ⓑ \(16{x}^{2}-40xy+25{y}^{2}\)
Multiply using the product of conjugates pattern: ⓐ \((2x+5)(2x-5)\) ⓑ \((5m-9n)(5m+9n).\)

Révèle la réponse

ⓐ

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

Révèle la réponse

ⓐ \(36{x}^{2}-25\)
ⓑ \(16{p}^{2}-49{q}^{2}\)
Multiply: ⓐ \((2x+7)(2x-7)\) ⓑ \((3x-y)(3x+y).\)

Révèle la réponse

ⓐ \(4{x}^{2}-49\) ⓑ \(9{x}^{2}-{y}^{2}\)
Choose the appropriate pattern and use it to find the product:
ⓐ \((2x-3)(2x+3)\) ⓑ \({(8x-5)}^{2}\) ⓒ \({(6m+7)}^{2}\) ⓓ \((5x-6)(6x+5).\)

Révèle la réponse

ⓐ \((2x-3)(2x+3)\)
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.
\(\begin{array}{llll} & & & \ (5x-6)(6x+5) \\ \text{Use FOIL.} & & & \ 30{x}^{2}+25x-36x-30 \\ \text{Simplify.} & & & \ 30{x}^{2}-11x-30\end{array}\)
Choose the appropriate pattern and use it to find the product:
ⓐ \((9b-2)(2b+9)\) ⓑ \({(9p-4)}^{2}\) ⓒ \({(7y+1)}^{2}\) ⓓ \((4r-3)(4r+3).\)

Révèle la réponse

ⓐ FOIL; \(18{b}^{2}+77b-18\)
ⓑ Binomial Squares; \(81{p}^{2}-72p+16\)
ⓒ Binomial Squares; \(49{y}^{2}+14y+1\)
ⓓ Product of Conjugates; \(16{r}^{2}-9\)
Choose the appropriate pattern and use it to find the product:
ⓐ \({(6x+7)}^{2}\) ⓑ \((3x-4)(3x+4)\) ⓒ \((2x-5)(5x-2)\) ⓓ \({(6n-1)}^{2}.\)

Révèle la réponse

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

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

Révèle la réponse

ⓐ

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

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

Révèle la réponse

ⓐ \((f\cdot g)(x)={x}^{3}-7{x}^{2}+13x-15\)
ⓑ \((f\cdot g)(2)=-9\)
For functions \(f(x)=x-7\) and \(g(x)={x}^{2}+8x+4,\) find ⓐ \((f\cdot g)(x)\) ⓑ \((f\cdot g)(2).\)

Révèle la réponse

ⓐ \((f\cdot g)(x)={x}^{3}+{x}^{2}-52x-28\)
ⓑ \((f\cdot g)(2)=-120\)
ⓐ \((6{y}^{7})(-3{y}^{4})\)
ⓑ \((\frac{4}{7}r{s}^{2})(14r{s}^{3})\)
ⓐ \((-10{x}^{5})(-3{x}^{3})\)
ⓑ \((\frac{5}{8}{x}^{3}y)(24{x}^{5}y)\)

Révèle la réponse

ⓐ \(30{x}^{8}\) ⓑ \(15{x}^{8}{y}^{2}\)
ⓐ \((-8{u}^{6})(-9u)\)
ⓑ \((\frac{2}{3}{x}^{2}y)(\frac{3}{4}x{y}^{2})\)
ⓐ \((-6{c}^{4})(-12c)\)
ⓑ \((\frac{3}{5}{m}^{3}{n}^{2})(\frac{5}{9}{m}^{2}{n}^{3})\)

Révèle la réponse

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

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Multiply Polynomials

Multiply monomials
Multiply a polynomial by a monomial
Multiply a binomial by a binomial
Multiply a polynomial by a polynomial
Multiply special products
Multiply polynomial functions
Distributive Property
FOIL Method

Questions people ask

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.

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Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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