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Multiply Polynomials

Multiply a polynomial by a monomial

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.

\(4(x+10)\)

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

u² + 5u

Try it.

\(q(q+7)\)

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.

\((2y-9)y\)

Solution

2y² − 9y

Try it.

\((8b-1)b\)

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.

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 the problem, review .

Die Antwort aufzeigen

\(2x+6\)
Distribute: \(-11(4-3a).\)
If you missed the problem, review .

Die Antwort aufzeigen

\(-44+33a\)
Combine like terms: \({x}^{2}+9x+7x+63.\)
If you missed the problem, review .

Die Antwort aufzeigen

\({x}^{2}+16x+63\)
Multiply: \(3(x+7).\)

Die Antwort aufzeigen

\(3(x+7)\)
Distribute.
\(3\cdot x+3\cdot 7\)
Simplify. \(3x+21\)
Multiply: \(6(x+8).\)

Die Antwort aufzeigen

6x + 48
Multiply: \(2(y+12).\)

Die Antwort aufzeigen

2y + 24
Multiply: \(x(x-8).\)

Die Antwort aufzeigen

Distribute.
Simplify.
Multiply: \(y(y-9).\)

Die Antwort aufzeigen

y² − 9y
Multiply: \(p(p-13).\)

Die Antwort aufzeigen

p² − 13p
Multiply: \(10x(4x+y).\)

Die Antwort aufzeigen

Distribute.
Simplify.
Multiply: \(8x(x+3y).\)

Die Antwort aufzeigen

8x² + 24xy
Multiply: \(3r(6r+s).\)

Die Antwort aufzeigen

18r² + 3rs
Multiply: \(-2x(5{x}^{2}+7x-3).\)

Die Antwort aufzeigen

\(-2x(5{x}^{2}+7x-3)\)
Distribute.
\(-2x⋅5{x}^{2}+(-2x)⋅7x-(-2x)⋅3\)
Simplify. \(-10{x}^{3}-14{x}^{2}+6x\)
Multiply: \(-4y(8{y}^{2}+5y-9).\)

Die Antwort aufzeigen

−32y³ − 20y² + 36y
Multiply: \(-6x(9{x}^{2}+x-1).\)

Die Antwort aufzeigen

−54x³ − 6x² + 6x
Multiply: \(4{y}^{3}({y}^{2}-8y+1).\)

Die Antwort aufzeigen

\(4{y}^{3}({y}^{2}-8y+1)\)
Distribute.
\(4{y}^{3}⋅{y}^{2}-4{y}^{3}⋅8y+4{y}^{3}⋅1\)
Simplify. \(4{y}^{5}-32{y}^{4}+4{y}^{3}\)
Multiply: \(3{x}^{2}(4{x}^{2}-3x+9).\)

Die Antwort aufzeigen

12x⁴ − 9x³ + 27x²
Multiply: \(8{y}^{2}(3{y}^{2}-2y-4).\)

Die Antwort aufzeigen

24y⁴ − 16y³ − 32y²
Multiply: \((x+3)p.\)

Die Antwort aufzeigen

\((x+3)p\)
Distribute.
\(x⋅p+3⋅p\)
Simplify. \(xp+3p\)
Multiply: \((x+8)p.\)

Die Antwort aufzeigen

xp + 8p
Multiply: \((a+4)p.\)

Die Antwort aufzeigen

ap + 4p
Multiply: \((x+6)(x+8).\)

Die Antwort aufzeigen

\((x+6)(x+8)\)
Distribute \((x+8)\).
Distribute again. \({x}^{2}+8x+6x+48\)
Simplify. \({x}^{2}+14x+48\)
Multiply: \((x+8)(x+9).\)

Die Antwort aufzeigen

x² + 17x + 72
Multiply: \((a+4)(a+5).\)

Die Antwort aufzeigen

a² + 9a + 20
Multiply: \((2x+9)(3x+4).\)

Die Antwort aufzeigen

\((2x+9)(3x+4)\)
Distribute. \((3x+4)\)
Distribute again. \(6{x}^{2}+8x+27x+36\)
Simplify. \(6{x}^{2}+35x+36\)
Multiply: \((5x+9)(4x+3).\)

Die Antwort aufzeigen

20x² + 51x + 27
Multiply: \((10m+9)(8m+7).\)

Die Antwort aufzeigen

80m² + 142m + 63
Multiply: \((4y+3)(6y-5).\)

Die Antwort aufzeigen

\((4y+3)(6y-5)\)
Distribute.
Distribute again. \(24{y}^{2}-20y+18y-15\)
Simplify. \(24{y}^{2}-2y-15\)
Multiply: \((7y+1)(8y-3).\)

Die Antwort aufzeigen

56y² − 13y − 3
Multiply: \((3x+2)(5x-8).\)

Die Antwort aufzeigen

15x² − 14x − 16
Multiply: \((x+2)(x-y).\)

Die Antwort aufzeigen

Distribute.
Distribute again.
Simplify. There are no like terms to combine.
Multiply: \((x+5)(x-y).\)

Die Antwort aufzeigen

x² − xy + 5x − 5y
Multiply: \((x+2y)(x-1).\)

Die Antwort aufzeigen

x² − x + 2xy − 2y
Multiply using the FOIL method: \((x+6)(x+9).\)

Die Antwort aufzeigen

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.
Multiply using the FOIL method: \((x+7)(x+8).\)

Die Antwort aufzeigen

x² + 15x + 56
Multiply using the FOIL method: \((y+14)(y+2).\)

Die Antwort aufzeigen

y² + 16y + 28
Multiply: \((y-8)(y+6).\)

Die Antwort aufzeigen

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

Die Antwort aufzeigen

y² + 5y − 24
Multiply: \((q-4)(q+5).\)

Die Antwort aufzeigen

q² + q − 20
Multiply: \((2a+3)(3a-1).\)

Die Antwort aufzeigen

Multiply the First terms.
Multiply the Outer terms.
Multiply the Inner terms.
Multiply the Last terms.
Combine like terms.

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.

\approx

approximately equal
Equal to the precision shown, not exactly.

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

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

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Multiply Polynomials

Multiply a polynomial by a monomial
Multiply a binomial by a binomial
Multiply a trinomial by a binomial
Multiply the
Multiply the
Multiply the
Multiply the
Combine like terms, when possible.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Versuch es selbst.

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

Mehr in Arithmetic

Order of operations Fractions Percentages Powers and roots

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.

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.



Multiply the First terms.
Multiply the Outer terms.
Multiply the Inner terms.
Multiply the Last terms.
Combine like terms.