MULTIPLY POLYNOMIALS
To find the product of two polynomials:
1) multiply each term of the first polynomial by each term of the second polynomial.
2) simplify, when possible, by combining like terms.
Examples:
1) 9y(12 - 4y + 5y2) =
2) (m - 3)(m - 9)
m(m - 9) - 3(m -9)
m2 - 9m - 3m + 27
m2 - 12m + 27
108y - 36y2 + 45y3
multiply each term in first by each term in second
collect like terms
NOTE: As you continue to multiply polynomials you may be able to skip writing this step,
m(m - 9) - 3(m -9), and mentally do the multiplication to get this step,m2 - 9m - 3m + 27.
3) (5x - 2y)(x + 7y)
5x(x +7y) - 2y(x + 7y)
5x2 + 35xy -2xy - 14y2
5x2 + 33xy - 14y2
4) (a - 3)(4a2 - 6a + 1)
a(4a2 - 6a + 1) - 3(4a2 - 6a + 1)
4a3 - 6a2 + a - 12a2 + 18a - 3
4a3 - 18a2 + 19a - 3
multiply each term in first by each term in second
collect like terms
multiply each term in first by each term in second
collect like terms
5)
(3x + 2y - 4)(x - 5y + 9)
3x(x - 5y + 9) + 2y(x - 5y +9) - 4(x - 5y + 9)
3x2 - 15xy + 27x +2xy -10y2 + 18 - 4x +20y - 36
3x2 - 13xy + 23x -10y2 - 18
6)
(a + 9)(a - 9)
a(a - 9) + 9(a - 9)
a2 - 9a + 9a - 81
a2 - 81
7) (n - 1/3)(n + 3/4)
n(n + 3/4) - 1/3(n +3/4)
n2 + 3/4n - 1/3n - 1/4
n2 + 5/12n - 1/4
(recall the steps to add unlike fractions)
8) (2x + 5)2
rewrite as (2x +5)(2x+5) and then proceed as the other problems
2x(2x + 5) + 5(2x +5)
4x2 + 10x + 10x + 25
4x2 + 20x + 25
9) (4m - n)3
rewrite as (4m - n)(4m - n)(4m - n)
multiply the first two terms: (4m - n)(4m - n)
= 16m2 - 8mn + n2
then take that product and multiply it by last term:
(16m2 - 8mn + n2)(4m - n)
16m2(4m - n) - 8mn(4m - n) + n2(4m - n)
64m3 - 16m2n - 32m2n - 8mn2 + 4mn2 - n3
64m3 - 48m2n - 4mn2 - n3
The binomials, (a + b) and (a – b) are called conjugates. The terms are the same but the
operation are opposites (one is a +, the other is – ).
When multiplying conjugates, you may multiply the first terms and the last terms to get the
product.
(a + b) (a – b) = a2 – b2
Examples:
10) (x – 5)(x + 5) = x2 – (5)2
= x2 – 25
11) (2x + 7)(2x – 7) = (2x)2 – (7)2
4x2 – 49
=
12) (3x – 8y)(3x + 8y)
= (3x)2 – (8y)2
= 9x2 – 64y2

13)  x 

2 
2
 x  
3 
3
=
2
x –  
3
2
2
=
x2 –
4
9