Multiplying polynomials

As shown in the example below, multiplying polynomials demand the following steps.

Use the distributive law of multiplication to multiply each term in the first polynomial by each term in the second polynomial. In this step, we multiply the coefficients and the variables separately. If the variables are the same, you can just add the exponents. If the variables are different, just put them next to each other.
Combine and add like terms to simplify the polynomial
Write the polynomial in standard form

How to multiply a monomial by a binomial

It is kind of easy to multiply a monomial by a binomial. Just multiply the monomial by each term in the binomial.

Example #1:

2x(5x + 1) = 2x(5x) + 2x(1)

2x(5x + 1) = 10x² + 2x

How to multiply a binomial by a binomial

Important concept

You must know what a term is when multiplying polynomials. It is because the goal is to multiply each term of the polynomial on the left by each term of the polynomial on the right and then adding the whole thing!

Example #2:

(2x³ + 8)(3x + 4)

It is quite common to use the foil method to multiply a binomial by a binomial. Suppose each binomial is written in standard form then the first term of 2x³ + 8 is 2x³ and the second term is 8. Here is how to follow foil to find the product of (2x³ + 8)(3x + 4).

Firsts:

Multiply the first term of the first binomial (binomial on the left) by the first term of the second binomial (binomial on the right)

2x³ × 3x = 6x⁴

Outers:

Multiply the first term of the first binomial by the second term of the second binomial

2x³ × 4 = 8x³

Inners:

Multiply the second term of the first binomial by the first term of the second binomial

8 × 3x = 24x

Lasts:

Multiply the second term of the first binomial by the second term of the second binomial

8 × 4 = 32

(2x³ + 8)(3x + 4) = 6x⁴ + 8x³ + 24x + 32

Check the lesson about about multiplying binomials to learn more

More examples about multiplying polynomials

How to multiply a trinomial by a trinomial

Example #3:

Multiply 4x³ + 2x + 5 by 3x⁴ + x + 6

(4x³ + 2x + 5) × (3x⁴ + x + 6)

The result of each multiplication or whatever will be added together is shown in bold.

The polynomial on the left is 4x³ + 2x + 5

Each term is separated by an addition sign.

The first term is 4x³

The second term is 2x

The third term is 5

The polynomial on the right is 3x⁴ + x + 6

Each term is separated by an addition sign.

The first term is 3x⁴

The second term is x

The third term is 6

Now multiply the first term of the polynomial on the left that is 4x³ by each term of the polynomial on the right and these are 3x⁴, x , and 6.

4x³ × 3x⁴ = 4 × 3 × x³ × x⁴ = 12x^{3 + 4} = 12x⁷

4x³ × x = 4 × x³ × x = 4 × x³ × x¹ = 4x^{3 + 1} = 4x⁴

4x³ × 6 = 4 × 6x³ = 24x³

Next, multiply the second term of the polynomial on the left that is 2x by each term of the polynomial on the right and these are 3x⁴, x , and 6.

2x × 3x⁴ = 2 × 3 × x × x⁴ = 2 × 3 × x¹ × x⁴ = 6x^{1 + 4} = 6x⁵

2x × x = 2 × x × x = 2 × x¹ × x¹ = 2x ^{1 + 1} = 2x²

2x × 6 = 2 × 6x = 12x

Finally, multiply the third term of the polynomial on the left that is 5 by each term of the polynomial on the right and these are 3x⁴, x , and 6.

5 × 3x⁴ = 15x⁴

5 × x = 5x

5 × 6 = 30

Adding the result in bold together, we get:

12x⁷ + 4x⁴ + 24x³ + 6x⁵ + 2x² + 12x + 15x⁴ + 5x + 30

Combine like terms

12x⁷ + (4x⁴ + 15x⁴) + 24x³ + 6x⁵ + 2x² + (12x + 5x) + 30

12x⁷ + 19x⁴ + 24x³ + 6x⁵ + 2x² + 17x + 30

Write in standard form

12x⁷ + 6x⁵ + 19x⁴ + 24x³ + 2x² + 17x + 30

Example #4:

Multiply 4x³ − 2x + 5 by 3x⁴ + x − 6

Example #4 is almost the same as example #3. We just incorporated a couple of subtraction signs.

My teaching experience has taught me that when multiplying polynomials, it is best to say to students to replace minus with + -

Then, do the exact same thing you did in example #3

(4x³ − 2x + 5) × (3x⁴ + x − 6) = (4x³ + -2x + 5) × (3x⁴ + x + -6)

4x³ × 3x⁴ = 12x⁷

4x³ × x = 4x⁴

4x³ × -6 = 4 × -6x³ = -24x³

Notice this time that the second term of the polynomial on the left has a negative next to it! Same thing for the third term of the polynomial on the right.

-2x × 3x⁴ = -2 × 3 × x × x⁴ = -2 × 3 × x¹ × x⁴ = -6x^{1 + 4} = -6x⁵

-2x × x = -2 × x × x = -2 × x¹ × x¹ = -2x^{1 + 1} = -2x²

-2x × -6 = -2 × -6x = 12x

5 × 3x⁴ = 15x⁴

5 × x = 5x

5 × -6 = -30

We get 12x⁷ + 4x⁴ + -24x³ + -6x⁵ + -2x² + 12x + 15x⁴ + 5x + -30

Combine like terms

12x⁷ + (4x⁴ + 15x⁴) + -24x³ + -6x⁵ + -2x² + (12x + 5x) + -30

12x⁷ + 19x⁴ + -24x³ + -6x⁵ + -2x² + 17x + -30

Write the polynomial in standard form

12x⁷ + -6x⁵ + 19x⁴ + -24x³ + -2x² + 17x + -30

Multiplying polynomials should be a breeze if you really understand the four examples above.