Polynomial Multiplication Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polynomial Multiplication.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Multiplying polynomials by distributing every term in one polynomial to every term in the other, then combining like terms.

Each term in the first polynomial must 'shake hands' with every term in the second. For two binomials like (x+3)(x+5)(x + 3)(x + 5), the FOIL method (First, Outer, Inner, Last) organizes the four handshakes: x⋅x+x⋅5+3⋅x+3⋅5x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Polynomial multiplication distributes each term of one factor across all terms of the other, then combines.

Common stuck point: The procedure for polynomial multiplication is the easy part; the trap is squaring a binomial as a2+b2a^2+b^2. Asking "Am I multiplying expressions so that every term in one meets every term in the other?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I multiplying expressions so that every term in one meets every term in the other?

Worked Examples

Example 1

easy
Multiply (x+4)(x+3)(x + 4)(x + 3) using FOIL.

Answer

x2+7x+12x^2 + 7x + 12

First step

1
Step 1: First: x⋅x=x2x \cdot x = x^2.

Full solution

  1. 2
    Step 2: Outer: x⋅3=3xx \cdot 3 = 3x. Inner: 4⋅x=4x4 \cdot x = 4x. Last: 4⋅3=124 \cdot 3 = 12.
  2. 3
    Step 3: Combine: x2+3x+4x+12=x2+7x+12x^2 + 3x + 4x + 12 = x^2 + 7x + 12.
  3. 4
    Check: At x=1x = 1: (5)(4)=20(5)(4) = 20 and 1+7+12=201 + 7 + 12 = 20 ✓
FOIL (First, Outer, Inner, Last) is a mnemonic for multiplying two binomials. Each term in the first binomial multiplies each term in the second.

Example 2

medium
Expand (2x−1)(x2+3x−5)(2x - 1)(x^2 + 3x - 5).

Example 3

medium
Multiply (3x+4)(2x−5)(3x + 4)(2x - 5) using FOIL.

Example 4

medium
A square has side length x+5x + 5. Express its area as an expanded polynomial.

Example 5

hard
A box has length x+4x + 4, width x+1x + 1, and height xx. Express the volume as an expanded polynomial.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Multiply (x−2)(x+6)(x - 2)(x + 6).

Example 2

hard
Expand (x+1)2(x−1)(x + 1)^2(x - 1).

Example 3

easy
Multiply x⋅x3x\cdot x^3.

Example 4

easy
Multiply 3x⋅4x3x\cdot4x.

Example 5

easy
Distribute 2(x+5)2(x+5).

Example 6

easy
Distribute x(x+3)x(x+3).

Example 7

easy
Multiply (x+2)(x+3)(x+2)(x+3) using FOIL.

Example 8

easy
Multiply (x−4)(x+4)(x-4)(x+4).

Example 9

easy
Multiply (x+1)(x+1)(x+1)(x+1).

Example 10

easy
Multiply −2x(3x−1)-2x(3x-1).

Example 11

medium
Multiply (2x+3)(x−5)(2x+3)(x-5).

Example 12

medium
Multiply (x+2)(x2−3x+1)(x+2)(x^2-3x+1).

Example 13

medium
Expand (x−3)2(x-3)^2.

Example 14

medium
Multiply (2x−1)(2x+1)(2x-1)(2x+1).

Example 15

medium
A rectangle has length x+5x+5 and width x−2x-2. Find its area as a polynomial.

Example 16

medium
Multiply 3x2(2x2−x+4)3x^2(2x^2-x+4).

Example 17

medium
Expand (x+y)(x−y+2)(x+y)(x-y+2).

Example 18

medium
Multiply (x+3)(x−3)(x+1)(x+3)(x-3)(x+1).

Example 19

medium
Multiply (3x−2)(3x+2)(3x-2)(3x+2).

Example 20

challenge
Find the coefficient of x2x^2 in (x+1)(x+2)(x+3)(x+1)(x+2)(x+3).

Example 21

challenge
Expand (x+2)3(x+2)^3 using polynomial multiplication.

Example 22

challenge
Show that (x+a)(x+b)=x2+(a+b)x+ab(x+a)(x+b)=x^2+(a+b)x+ab and use it to expand (x+7)(x−2)(x+7)(x-2).

Example 23

easy
Multiply 5x2⋅4x35x^2 \cdot 4x^3.

Example 24

easy
Distribute 5(2x−3)5(2x - 3).

Example 25

easy
Distribute −3x(2x2−x+4)-3x(2x^2 - x + 4).

Example 26

easy
Expand (x+6)2(x + 6)^2.

Example 27

medium
Multiply (x−4)(x2+2x−3)(x - 4)(x^2 + 2x - 3).

Example 28

medium
Multiply (2x+1)(2x−1)(x+3)(2x + 1)(2x - 1)(x + 3).

Example 29

medium
Find the coefficient of xx in the expansion of (x+3)(x−5)(x + 3)(x - 5).

Example 30

medium
Expand (x+2y)(x−2y)(x + 2y)(x - 2y).

Example 31

medium
Multiply 4x2(3x3−2x+7)4x^2(3x^3 - 2x + 7).

Example 32

medium
Multiply (x2+1)(x2−1)(x^2 + 1)(x^2 - 1).

Example 33

medium
Multiply (x+1)(x+2)(x+3)(x+4)(x + 1)(x + 2)(x + 3)(x + 4) and find the coefficient of x3x^3.

Example 34

medium
Multiply (x+a)2−(x−a)2(x + a)^2 - (x - a)^2 and simplify.

Example 35

hard
Expand (x+2)3(x + 2)^3 using polynomial multiplication.

Example 36

hard
Multiply (x2+x+1)(x2−x+1)(x^2 + x + 1)(x^2 - x + 1).

Example 37

hard
Find the coefficient of x2x^2 in (x+1)4(x + 1)^4.

Example 38

hard
Multiply (x+2)(x−3)(x+5)(x + 2)(x - 3)(x + 5) and write in standard form.

Example 39

hard
Expand (2x−1)3(2x - 1)^3.

Example 40

challenge
Find the coefficient of x3x^3 in the expansion of (1+x+x2)3(1 + x + x^2)^3.

Example 41

challenge
If (x+a)(x+b)=x2+11x+24(x + a)(x + b) = x^2 + 11x + 24, find a+ba + b and a⋅ba \cdot b.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

polynomialsdistributive property