Math Core

Adding polynomials only needed like terms. Multiplying them needs two tools you already have: the product rule for exponents and the distributive property. Put together, they let you multiply any two polynomials, which is the skill that factoring will later run in reverse.

Multiplying monomials

To multiply monomials, multiply the coefficients and use the product rule, $x^m \cdot x^n = x^{m+n}$ , for each variable.

(-3x^4)(5x^2) = (-3 \cdot 5)(x^4 \cdot x^2) = -15x^6

(2a^2b)(6ab^3) = (2 \cdot 6)(a^2 \cdot a)(b \cdot b^3) = 12a^3b^4

Remember that a variable with no written exponent has exponent $1$ , so $a^2 \cdot a = a^3$ .

A monomial times a polynomial

Distribute the monomial to each term of the polynomial, multiplying monomials each time.

Worked example: Distributing a monomial

Find $3x^2(4x^3 - 2x + 5)$ .

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

A binomial times a binomial

To multiply $(x + 4)(x - 6)$ , treat the first binomial as a single quantity and distribute it over the second. Then distribute again:

\begin{aligned} (x + 4)(x - 6) &= (x + 4)\cdot x + (x + 4)\cdot(-6) \\ &= x^2 + 4x - 6x - 24 \\ &= x^2 - 2x - 24 \end{aligned}

The upshot: every term in the first factor multiplies every term in the second. Two terms times two terms gives four products, which you then combine.

The area model

A rectangle with side lengths $x + 4$ and $x - 6$ splits into four smaller rectangles, one for each pair of terms. A table keeps them organized:

$\times$	$x$	$-6$
$x$	$x^2$	$-6x$
$4$	$4x$	$-24$

Add the four cells and combine like terms: $x^2 - 6x + 4x - 24 = x^2 - 2x - 24$ . The like terms always land on a diagonal of the table, which makes them easy to spot.

FOIL

For two binomials, some people remember the four products with FOIL: First, Outer, Inner, Last.

Worked example: Using FOIL

Find $(2x - 3)(3x + 5)$ .

First: $2x \cdot 3x = 6x^2$
Outer: $2x \cdot 5 = 10x$
Inner: $-3 \cdot 3x = -9x$
Last: $-3 \cdot 5 = -15$

Add them: $6x^2 + 10x - 9x - 15 = 6x^2 + x - 15$ .

FOIL is just the distributive property with a checklist. It only works for binomial times binomial; for anything bigger, use distribution or a table.

Multiplying polynomials

To multiply two polynomials, multiply each term of the first by each term of the second, then combine like terms.

If the factors have $m$ and $n$ terms, there are $m \times n$ products before combining. The degree of the product is the sum of the degrees of the factors.

Common mistake

Don't multiply only the "matching" terms. $(x + 4)(x - 6)$ is not $x^2 - 24$ ; that skips the outer and inner products $-6x$ and $4x$ . Always count your products: two binomials must give four of them.

Larger products

The same idea handles a binomial times a trinomial. There are $2 \times 3 = 6$ products.

Worked example: Binomial times trinomial

Find $(x + 2)(x^2 - 3x + 4)$ .

Distribute $x$ , then distribute $2$ :

\begin{aligned} &= x(x^2 - 3x + 4) + 2(x^2 - 3x + 4) \\ &= x^3 - 3x^2 + 4x + 2x^2 - 6x + 8 \\ &= x^3 - x^2 - 2x + 8 \end{aligned}

The table version:

$\times$	$x^2$	$-3x$	$4$
$x$	$x^3$	$-3x^2$	$4x$
$2$	$2x^2$	$-6x$	$8$

The $x^2$ -terms $-3x^2$ and $2x^2$ combine to $-x^2$ ; the $x$ -terms $4x$ and $-6x$ combine to $-2x$ .

Worked example: An area

A rectangular patio is $2x + 3$ meters long and $x - 1$ meters wide. Write its area as a polynomial, then find the area when $x = 5$ .

(2x + 3)(x - 1) = 2x^2 - 2x + 3x - 3 = 2x^2 + x - 3.

When $x = 5$ : $2(25) + 5 - 3 = 52$ square meters. Check with the factored form: the patio is $13$ by $4$ , and $13 \cdot 4 = 52$ . ✓

Tip

Before multiplying, predict the leading term and the constant. The leading term is the product of the two leading terms, and the constant is the product of the two constants. In $(2x + 3)(x - 1)$ , you should get $2x^2$ first and $-3$ last. If your answer doesn't start and end that way, something slipped.

Practice

Practice 1

Find $(-2a^3)(5a^4)$ .

Practice 2

Find $4x(3x^2 - x + 2)$ .

Practice 3

When $(x + 7)(x - 3)$ is multiplied out and simplified, what is the coefficient of $x$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Find $(2y + 1)(4y - 3)$ .

Practice 5

When $(3x - 2)(2x - 5)$ is multiplied out and simplified, what is the coefficient of $x$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

When $(x - 1)(x^2 + 2x - 5)$ is multiplied out and simplified, what is the coefficient of $x$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

What is the degree of $(x^3 + 1)(2x^4 - x)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A rectangle has length $2x + 3$ and width $x - 4$ . Which polynomial gives its area?