A binomial is a polynomial with two terms. For example, x, minus, 2 and x, minus, 6 are both binomials. In this article, we'll review how to multiply these binomials.

Example 1

Expand the expression.
left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, minus, 6, right parenthesis
Apply the distributive property.
(x2)(x6)=x(x6)2(x6)\begin{aligned}&(\blueD{x-2})(x-6)\\ \\ =&\blueD{x}(x-6)\blueD{-2}(x-6)\\ \end{aligned}
Apply the distributive property again.
equals, start color blueD, x, end color blueD, left parenthesis, x, right parenthesis, plus, start color blueD, x, end color blueD, left parenthesis, minus, 6, right parenthesis, start color blueD, minus, 2, end color blueD, left parenthesis, x, right parenthesis, start color blueD, minus, 2, end color blueD, left parenthesis, minus, 6, right parenthesis
Notice the pattern. We multiplied each term in the first binomial by each term in the second binomial.
Simplify.
=x26x2x+12=x28x+12\begin{aligned} =&x^2-6x-2x+12\\\\ =&x^2-8x+12 \end{aligned}

Example 2

Expand the expression.
left parenthesis, minus, a, plus, 1, right parenthesis, left parenthesis, 5, a, plus, 6, right parenthesis
Apply the distributive property.
(a+1)(5a+6)=a(5a+6)+1(5a+6)\begin{aligned} &(\purpleD{-a+1})(5a+6)\\\\ =&\purpleD{-a}(5a+6) +\purpleD{1}(5a+6) \end{aligned}
Apply the distributive property again.
equals, start color purpleD, minus, a, end color purpleD, left parenthesis, 5, a, right parenthesis, start color purpleD, minus, a, end color purpleD, left parenthesis, 6, right parenthesis, plus, start color purpleD, 1, end color purpleD, left parenthesis, 5, a, right parenthesis, plus, start color purpleD, 1, end color purpleD, left parenthesis, 6, right parenthesis
Notice the pattern. We multiplied each term in the first binomial by each term in the second binomial.
Simplify:
minus, 5, a, start superscript, 2, end superscript, minus, a, plus, 6
Want to learn more about multiplying binomials? Check out this video.

Practice

Problem 1
Simplify.
Express your answer as a quadratic in standard form.
left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 6, right parenthesis

Want more practice? Check out this intro exercise and this slightly harder exercise.