Multiply Polynomials

How to multiply polynomials (binomial times trinomial, trinomial times trinomial): 2 examples and their solutions.

Example 1

Example

Solution

Multiply x and each term of (x2 - x + 7).
x⋅x2 = x3
x⋅(-x) = -x2
x⋅(+7) = +7x

Multiply a Monomial and a Polynomial

Multiply -3 and each term of (x2 - x + 7).

-3⋅x2 = -3x2
Write this under -x2.

-3⋅(-x) = +3x
Write this under +7x.

-3⋅(+7) = -21
Write this behind +3x.

So (x - 3)(x2 - x + 7)
= x3 - x2 + 7x - 3x2 + 3x - 21.

Combine the like terms.
-x2 - 3x2 = -4x2
+7x + 3x = +10x

Add and Subtract Polynomials

So x3 - 4x2 + 10x - 21 is the answer.

Example 2

Example

Solution

Multiply x2 and each term of (3x2 - x + 8).
x2⋅3x2 = 3x4
x2⋅(-x) = -x3
x2⋅(+8) = +8x2

Multiply +7x and each term of (3x2 - x + 8).

+7x⋅3x2 = +21x3
Write this under -x3.

+7x⋅(-x) = -7x2
Write this under +8x2.

+7x⋅(+8) = +56x
Write this behind -7x2.

Multiply -1 and each term of (3x2 - x + 8).

-1⋅3x2 = -3x2
Write this under -7x2.

-1⋅(-x) = +x
Write this under +56x.

-1⋅(+8) = -8
Write this behind +x.

So (x2 + 7x - 1)(3x2 - x + 8)
= 3x4 - x3 + 8x2 + 21x3 - 7x2 + 56x - 3x2 + x - 8.

Combine the like terms.
-x3 + 21x3 = +20x3
+8x2 - 7x2 - 3x2 = (+8 - 7 - 3)x2 = -2x2
+56x + x = +57x

So 3x4 + 20x3 - 2x2 + 57x - 8 is the answer.