Introduction

When we want to multiply two binomials, we take each term of the first binomial and multiply it by the second binomial. Then, we simplify by collecting like terms.

For example, to multiply ({\color{blue}{x}} + {\color{red}{3}}) and (x + 2), we multiply {\color{blue}{x}} and (x + 2), and then we add this to the product of {\color{red}{3}} and (x+2) :

\eqalign{ ({\color{blue}{x}} + {\color{red}{3}})(x + 2) &= \\[5pt] {\color{blue}{x}}(x + 2) + {\color{red}{3}}(x + 2) &= \\[5pt] x^2 + 2x + 3x + 6 &= \\[5pt] x^2 + 5x + 6 }

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Example: Multiplying Binomials

Expand and simplify (y + 5)(y - 2).

EXPLANATION

We take each term of the first binomial and multiply it by the second binomial. Then, we simplify by collecting like terms.

\eqalign{ (y + 5)(y - 2) &= \\ y(y-2) + 5(y-2) &=\\ y^2-2y +5y-10 &=\\ y^2 + 3y - 10 }

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Practice: Multiplying Binomials

$(q + 6)(q + 2) =$

a
 $q+8$
b
 $9q+12$
c
 $q^2 +8q + 12$
d
 $q^2+12$
e
 $9q^2+12$
Practice: Multiplying Binomials

Expand the following product of binomials. Write your answer in standard form.

a
b
c
d
e
Example: Multiplying Binomials with Leading Coefficients

Expand and simplify (2n - 1)(3n - 6).

EXPLANATION

We take each term of the first binomial and multiply it by the second binomial. Then, we simplify by collecting like terms.

\eqalign{ (2n - 1)(3n - 6) &= \\ 2n(3n - 6) + (-1)(3n - 6) &= \\ 6n^2 - 12n - 3n + 6 &= \\ 6n^2 - 15n + 6 }

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Practice: Multiplying Binomials with Leading Coefficients

Expand the following product of binomials. Write your answer in standard form.

a
b
c
d
e
Practice: Multiplying Binomials with Leading Coefficients

$(3x + 2)(5x - 4)=$

a
 $15x^2 -2x - 8$
b
 $15x^2-8$
c
 $15x^2+2x+8$
d
 $15x^2 +22x - 8$
e
 $15x^2+2x-8$
Example: Multiplying Binomials with Higher Powers and Multiple Variables

Expand (p^2-q^2)(2p+3q).

EXPLANATION

We take each term of the first binomial and multiply it by the second binomial. Then, we simplify by collecting like terms.

\begin{align*} (p^2-q^2)(2p+3q) &=\\ p^2(2p+3q) + (-q^2)(2p+3q) &=\\ 2p^3 +3p^2q - 2pq^2 -3q^3 & \end{align*}

We cannot simplify the above expression any further because there are no like terms.

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Practice: Multiplying Binomials with Higher Powers and Multiple Variables

$(5a+b)(a+3b)=$

a
 $a^2+14ab+3b^2$
b
 $5a^2+16ab+3b^2$
c
 $5a^2+12ab+b^2$
d
 $5a^2+14ab+3b^2$
e
 $a^2+16ab+b^2$
Practice: Multiplying Binomials with Higher Powers and Multiple Variables

$(7u^2-6v)(v-u) =$

a
b
c
d
e
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