# 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 (x^{2} - x + 7).

x⋅x^{2} = x^{3}

x⋅(-x) = -x^{2}

x⋅(+7) = +7x

Multiply a Monomial and a Polynomial

Multiply -3 and each term of (x^{2} - x + 7).

-3⋅x^{2} = -3x^{2}

Write this under -x^{2}.

-3⋅(-x) = +3x

Write this under +7x.

-3⋅(+7) = -21

Write this behind +3x.

So (x - 3)(x^{2} - x + 7)

= x^{3} - x^{2} + 7x - 3x^{2} + 3x - 21.

Combine the like terms.

-x^{2} - 3x^{2} = -4x^{2}

+7x + 3x = +10x

Add and Subtract Polynomials

So x^{3} - 4x^{2} + 10x - 21 is the answer.

## Example 2

### Example

### Solution

Multiply x^{2} and each term of (3x^{2} - x + 8).

x^{2}⋅3x^{2} = 3x^{4}

x^{2}⋅(-x) = -x^{3}

x^{2}⋅(+8) = +8x^{2}

Multiply +7x and each term of (3x^{2} - x + 8).

+7x⋅3x^{2} = +21x^{3}

Write this under -x^{3}.

+7x⋅(-x) = -7x^{2}

Write this under +8x^{2}.

+7x⋅(+8) = +56x

Write this behind -7x^{2}.

Multiply -1 and each term of (3x^{2} - x + 8).

-1⋅3x^{2} = -3x^{2}

Write this under -7x^{2}.

-1⋅(-x) = +x

Write this under +56x.

-1⋅(+8) = -8

Write this behind +x.

So (x^{2} + 7x - 1)(3x^{2} - x + 8)

= 3x^{4} - x^{3} + 8x^{2} + 21x^{3} - 7x^{2} + 56x - 3x^{2} + x - 8.

Combine the like terms.

-x^{3} + 21x^{3} = +20x^{3}

+8x^{2} - 7x^{2} - 3x^{2} = (+8 - 7 - 3)x^{2} = -2x^{2}

+56x + x = +57x

So 3x^{4} + 20x^{3} - 2x^{2} + 57x - 8 is the answer.