Multiplying Polynomials
To multiply a monomial by a polynomial, we simply distribute the monomial.
3𝑥(2𝑥 2 + 4𝑥 − 7)
= 6𝑥 3 + 12𝑥 2 − 21𝑥
To multiply a binomial by another binomial, we often use the FOIL acronym (First Outer Inner
Last).
(5𝑥 + 9)(2𝑥 − 11)
= 10𝑥 2 − 55𝑥 + 18𝑥 − 99
= 10𝑥 2 − 37𝑥 − 99
For polynomials with more than two terms, FOIL does not work because it leaves out at least one
term. In these cases, we use repeated distribution. That is, we distribute all the terms in one
polynomial through the other polynomial and then combine like terms. (This is really what we
are doing when we use FOIL; we just don’t always think of it that way.)
(4𝑥 − 3)(𝑥 2 − 5𝑥 + 2)
= 4𝑥(𝑥 2 − 5𝑥 + 2) − 3(𝑥 2 − 5𝑥 + 2)
= 4𝑥 3 − 20𝑥 2 + 8𝑥 − 3𝑥 2 + 15𝑥 − 6
= 4𝑥 3 − 23𝑥 2 + 23𝑥 − 6
We don’t always write this step.
I’m showing it here to explain
where the terms in the next step
came from.
Some people prefer to use a vertical format when multiplying polynomials, much like we used in
earlier years to multiply longer numbers by hand. The idea is to line up like terms in columns as
you perform the distribution.
(5𝑥 + 4)(𝑥 2 + 2𝑥 − 3)
𝑥2
×
+5𝑥 3
5𝑥 3
+4𝑥 2
+10𝑥 2
+14𝑥 2
+2𝑥
5𝑥
+8𝑥
−15𝑥
−7𝑥
−3
+4
−12
−12
One benefit to using the vertical format is that you are more likely to catch a mistake with
exponents as it would disrupt the visual pattern. But it’s up to you whether you want to use this
format or not. As long as you are careful, you should get the same result using either format.
To multiply three polynomials, first multiply two of them. Then multiply that result by the third.
(𝑥 + 3)(𝑥 + 2)(𝑥 + 10)
= (𝑥 2 + 5𝑥 + 6)(𝑥 + 10)
First we multiply (𝑥 + 3) by (𝑥 + 2)
to get (𝑥 2 + 5𝑥 + 6).
= 𝑥 3 + 15𝑥 2 + 56𝑥 + 60
Then we multiply (𝑥 2 + 5𝑥 + 6) by
(𝑥 + 10) to get the final product.
To raise a polynomial to a power, we multiply it by itself the appropriate number of times. Note
that we do not simply apply the exponent to the individual terms.
(𝑥 2 + 4𝑥 − 7)2
𝑥4
+4𝑥 3
+4𝑥 3
𝑥4
+8𝑥 3
(𝑥 + 5)3


(𝑥 2 + 4𝑥 − 7)(𝑥 2 + 4𝑥 − 7)
−7𝑥 2
+16𝑥 2 −28𝑥
−7𝑥 2 −28𝑥
+2𝑥 2 −56𝑥
+49
+49
(𝑥 + 5)(𝑥 + 5)(𝑥 + 5)
Multiply the first two binomials:
(𝑥 + 5)(𝑥 + 5) = (𝑥 2 + 10𝑥 + 5)
Then multiply the result by the third binomial:
𝑥3
𝑥3
+5𝑥 2
+10𝑥 2
+15𝑥 2
+50𝑥
+5𝑥
+55𝑥
+25
+25
(𝑥 2 + 10𝑥 + 5)(𝑥 + 5)