Multiplying Polynomials

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Multiplying polynomials means multiplying every term of one polynomial by every term of the other, then combining like terms. You can use the distributive property or a box (grid) to keep track of the products, and count them so you don’t miss one. For example, (x + 2)(x^2 + 3x - 1) = x^3 + 5x^2 + 5x - 2.

Before this lesson: You should know multiplying binomials, the exponent rules, and combining like terms.

How Multiplying Polynomials Works

When you multiplied two binomials, you made 4 products. The same idea works for any polynomials: every term must meet every other term once.

So you can count the products before you start. A binomial times a trinomial gives 2 \cdot 3 = 6 products. A trinomial times a trinomial gives 3 \cdot 3 = 9. If you end up with fewer, you missed one.

Each product uses the product rule for exponents: multiply the coefficients and add the exponents. For example, (-4x)(2x) = -8x^2.

A 2 by 3 grid. Across the top: x², −4x and 5. Down the side: 2x and −3. The boxes are 2x³, −8x², 10x in the first row and −3x², 12x, −15 in the second row. Like terms share a color and border style: the two x² boxes are orange and dashed, the two x boxes are green and dotted. The answer is 2x³ − 11x² + 22x − 15.
Each box is one product. Like terms sit on diagonals, so they are easy to find and combine.

Rules for Multiplying Polynomials

TypeRuleExample
Monomial × polynomialDistribute the monomial to every term.2x(x^2 + 3) = 2x^3 + 6x
Binomial × binomial4 products (FOIL works here).(x + 1)(x + 4) = x^2 + 5x + 4
Any polynomial × polynomialEvery term times every term, then combine.(x + 1)(x^2 - x + 1) = x^3 + 1

A handy fact: the degree of the product is the sum of the degrees. A degree-1 polynomial times a degree-2 polynomial gives degree 3.

Steps

  1. Count the terms in each polynomial. Multiply the counts to know how many products to expect.
  2. Take the first term of the first polynomial and multiply it by every term of the second.
  3. Repeat with each remaining term of the first polynomial. Keep each sign with its term.
  4. Combine like terms.
  5. Write the answer in standard form, from the highest power down.

Worked Examples

Example 1: Monomial times a trinomial

Multiply 3x^2(2x^2 - 5x + 4).

\begin{aligned} 3x^2(2x^2 - 5x + 4) &= 3x^2 \cdot 2x^2 + 3x^2 \cdot (-5x) + 3x^2 \cdot 4 \\ &= 6x^4 - 15x^3 + 12x^2 \end{aligned}

There are no like terms to combine, because each product has a different power.

Example 2: Binomial times a trinomial

Multiply (x + 2)(x^2 + 3x - 1). Expect 2 \cdot 3 = 6 products.

\begin{aligned} &= x(x^2 + 3x - 1) + 2(x^2 + 3x - 1) \\ &= x^3 + 3x^2 - x + 2x^2 + 6x - 2 \\ &= x^3 + 5x^2 + 5x - 2 \end{aligned}

Check with x = 1: (3)(3) = 9 and 1 + 5 + 5 - 2 = 9.

Example 3: The box method

Multiply (2x - 3)(x^2 - 4x + 5).

×x^2-4x5
2x2x^3-8x^210x
-3-3x^212x-15

Combine the like terms: -8x^2 - 3x^2 = -11x^2 and 10x + 12x = 22x. The answer is 2x^3 - 11x^2 + 22x - 15.

Notice (-3)(-4x) = 12x. A negative times a negative is positive.

Example 4: Trinomial times a trinomial

Multiply (x^2 + 2x - 1)(x^2 - 3x + 2). Expect 3 \cdot 3 = 9 products.

\begin{aligned} &= x^2(x^2 - 3x + 2) + 2x(x^2 - 3x + 2) - 1(x^2 - 3x + 2) \\ &= x^4 - 3x^3 + 2x^2 + 2x^3 - 6x^2 + 4x - x^2 + 3x - 2 \\ &= x^4 - x^3 - 5x^2 + 7x - 2 \end{aligned}

Group by power to combine: x^3: -3 + 2 = -1. x^2: 2 - 6 - 1 = -5. x: 4 + 3 = 7.

Example 5: Word problem with three factors

A box is x inches tall, x + 2 inches wide and x + 5 inches long. Write its volume as a polynomial. Then find the volume when x = 3.

Multiply two factors first, then multiply by the third.

\begin{aligned} V &= x(x + 2)(x + 5) \\ &= x(x^2 + 7x + 10) \\ &= x^3 + 7x^2 + 10x \end{aligned}

When x = 3: V = 27 + 63 + 30 = 120 cubic inches. Check: the box is 3 by 5 by 8 inches, and 3 \cdot 5 \cdot 8 = 120.

Common Mistakes

MistakeExample of the errorFix
Missing products(x + 2)(x^2 + 3x - 1) with only 4 productsA binomial times a trinomial needs 6 products. Count them.
Forgetting the exponent on xx \cdot x^2 = x^2Add the exponents: x^1 \cdot x^2 = x^3.
Multiplying the exponentsx^2 \cdot x^3 = x^6Add them: x^2 \cdot x^3 = x^5.
Dropping a negative(-3)(-4x) = -12xA negative times a negative is positive: 12x.
Cubing each term(x + 1)^3 = x^3 + 1Multiply it out: (x + 1)^3 = x^3 + 3x^2 + 3x + 1.

The Vertical Method and Checking Your Work

You can also multiply polynomials like multi-digit numbers. Write one polynomial above the other. Multiply by each bottom term on its own row, and line up like terms in columns. Then add the columns.

x^3x^2xConstant
2 \cdot (x^2 + 3x - 1)2x^26x-2
x \cdot (x^2 + 3x - 1)x^33x^2-x
Sumx^35x^25x-2

This is Example 2 again, and it gives the same answer. To check any product, put a small number like x = 1 or x = 2 into the problem and into your answer. The two results must match, just like when adding and subtracting polynomials.

Practice Problems

  1. Multiply 4x(3x^2 - 2x + 7).
  2. Multiply -2y^2(y^3 - 4y + 1).
  3. Multiply (x + 4)(x^2 - x + 3).
  4. Multiply (x - 1)(x^2 + x + 1).
  5. Multiply (3x + 2)(2x^2 - x - 4).
  6. Multiply (a - 2b)(a^2 + 3ab - b^2).
  7. Multiply (x^2 + 2)(x^2 - 3x + 1).
  8. Multiply (x + 1)^3.
  9. Without multiplying everything out, find the degree and leading coefficient of (2x^3 + x - 5)(x^4 - 7x^2 + 1).
  10. A box is x feet tall, x - 1 feet wide and x + 4 feet long. Write its volume as a polynomial, and find the volume when x = 5.
Show answers
  1. 4x \cdot 3x^2 + 4x \cdot (-2x) + 4x \cdot 7 = 12x^3 - 8x^2 + 28x
  2. -2y^5 + 8y^3 - 2y^2. Each sign flips because the monomial is negative.
  3. x^3 - x^2 + 3x + 4x^2 - 4x + 12 = x^3 + 3x^2 - x + 12
  4. x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1. The middle terms all cancel.
  5. 6x^3 - 3x^2 - 12x + 4x^2 - 2x - 8 = 6x^3 + x^2 - 14x - 8
  6. a^3 + 3a^2b - ab^2 - 2a^2b - 6ab^2 + 2b^3 = a^3 + a^2b - 7ab^2 + 2b^3
  7. x^4 - 3x^3 + x^2 + 2x^2 - 6x + 2 = x^4 - 3x^3 + 3x^2 - 6x + 2
  8. (x + 1)(x + 1) = x^2 + 2x + 1. Then (x + 1)(x^2 + 2x + 1) = x^3 + 2x^2 + x + x^2 + 2x + 1 = x^3 + 3x^2 + 3x + 1.
  9. Degree 3 + 4 = 7. The leading term is 2x^3 \cdot x^4 = 2x^7, so the leading coefficient is 2.
  10. x(x - 1)(x + 4) = x(x^2 + 3x - 4) = x^3 + 3x^2 - 4x. At x = 5: 125 + 75 - 20 = 180 cubic feet. Check: 5 \cdot 4 \cdot 9 = 180.

FAQ

How do you multiply polynomials?

Multiply each term of the first polynomial by each term of the second. Then combine like terms and write the answer in standard form.

How many terms do you get before combining?

Multiply the number of terms in each polynomial. A binomial times a trinomial gives 6 terms before you combine like terms.

Can you use FOIL to multiply polynomials?

Only when both are binomials. For anything bigger, use the distributive property, the box method or the vertical method.

How do you multiply three polynomials?

Multiply two of them first. Then multiply that answer by the third one. For example, x(x + 2)(x + 5) = x(x^2 + 7x + 10).

Summary

  • Multiply every term of one polynomial by every term of the other.
  • The number of products equals the number of terms in the first times the number in the second.
  • Multiply coefficients and add exponents for each product.
  • Combine like terms and write the answer in standard form.
  • The box and vertical methods help you keep track of terms and signs.

Previous lesson: Multiplying Binomials

Next lesson: Special Products of Binomials (coming soon)

Back to the Algebra 1 Index

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