You probably know how to multiply monomials. For example,
-2x3 Â· 4x2 = -8x5.
To multiply a monomial and a polynomial of two or more terms, we apply the
distributive property. For example,
3x(x3 - 5) = 3x4 - 15x.
Multiplying a monomial and a polynomial
Find the products.
a) 2ab2 Â· 3a2b
b) (-1)(5 - x)
c) (x3 - 5x + 2)(-3x)
a) 2ab2 Â· 3a2b = 6a3b3
b) (-1)(5 - x) = -5 + x = x - 5
c) Each term of x3 - 5x + 2 is multiplied by -3x:
(x3 - 5x + 2)(-3x) = -3x4 + 15x2 - 6x
Note what happened to the binomial in Example 1(b) when we multiplied it
by -1. If we multiply any difference by -1, we get the same type of result:
-1(a - b) = -a + b = b - a.
Because multiplying by -1 is the same as taking the opposite, we can write this
-(a - b) = b - a.
This equation says that a - b and b - a are opposites or additive inverses of each
other. Note that the opposite of a b is -a - b, not a - b.
To multiply a binomial and a trinomial, we can use the distributive property or
set it up like multiplication of whole numbers.
Multiplying a binomial and a trinomial
Find the product (x + 2)(x2 + 3x - 5).
We can find this product by applying the distributive property twice. First we
multiply the binomial and each term of the trinomial:
|(x + 2)(x2 + 3x - 5)
||= (x + 2)x2 + (x + 2)3x + (x + 2)(-5)
||= x3 + 2x2 + 3x2 + 6x -
5x - 10
||= x3 + 5x2 + x - 10
||Combine like terms.
We could have found this product vertically:
||+ 3x -
|| 2(x2 + 3x - 5) = 2x2
+ 6x - 10
|| x(x2 + 3x - 5) = x3
+ 3x2 - 5x
||+ x -
Many students find vertical
multiplication easier than applying
the distributive property
twice horizontally. However,
you should learn both
methods because horizontal
multiplication will help you
with factoring by grouping.