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

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) = 10x^{2} + 2x

**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:**

(2x^{3} + 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 2x^{3} + 8 is 2x^{3} and the second term is 8. Here is how to follow foil to find the product of (2x^{3} + 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)

2x^{3} × 3x = 6x^{4}

**Outers:**

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

2x^{3} × 4 = 8x^{3}

**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

(2x^{3} + 8)(3x + 4) = 6x^{4} + 8x^{3} + 24x + 32

Check the lesson about about multiplying binomials to learn more

**Example #3:**

Multiply 4x^{3} + 2x + 5 by 3x^{4} + x + 6

(4x^{3} + 2x + 5) × (3x^{4} + x + 6)

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

The polynomial on the left is 4x^{3} + 2x + 5

Each term is separated by an addition sign.

- The first term is 4x
^{3}

- The second term is 2x

- The third term is 5

The polynomial on the right is 3x^{4} + x + 6

Each term is separated by an addition sign.

- The first term is 3x
^{4}

- The second term is x

- The third term is 6

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

4x^{3} × 3x^{4} = 4 × 3 × x^{3} × x^{4} = 12x^{3 + 4} = **12x ^{7}**

4x^{3} × x = 4 × x^{3} × x = 4 × x^{3} × x^{1} = 4x^{3 + 1} = **4x ^{4}**

4x^{3} × 6 = 4 × 6x^{3} = **24x ^{3}**

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 3x^{4}, x , and 6.

2x × 3x^{4} = 2 × 3 × x × x^{4} = 2 × 3 × x^{1} × x^{4} = 6x^{1 + 4} = **6x ^{5}**

2x × x = 2 × x × x = 2 × x^{1} × x^{1} = 2x ^{ 1 + 1} = **2x ^{2}**

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 3x^{4}, x , and 6.

5 × 3x^{4} = **15x ^{4}**

5 × x = **5x**

5 × 6 = **30**

Adding the result in bold together, we get:

12x^{7} + 4x^{4} + 24x^{3} + 6x^{5} + 2x^{2} + 12x + 15x^{4} + 5x + 30

Combine like terms

12x^{7} + (4x^{4} + 15x^{4}) + 24x^{3} + 6x^{5} + 2x^{2} + (12x + 5x) + 30

12x^{7} + 19x^{4} + 24x^{3} + 6x^{5} + 2x^{2} + 17x + 30

Write in standard form

12x^{7} + 6x^{5} + 19x^{4} + 24x^{3} + 2x^{2} + 17x + 30

**Example #4:**

Multiply 4x^{3} − 2x + 5 by 3x^{4} + 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**

(4x^{3} − 2x + 5) × (3x^{4} + x − 6) = (4x^{3} + -2x + 5) × (3x^{4} + x + -6)

4x^{3} × 3x^{4} = 12x^{7}

4x^{3} × x = 4x^{4}

4x^{3} × -6 = 4 × -6x^{3} = -24x^{3}

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 × 3x^{4} = -2 × 3 × x × x^{4} = -2 × 3 × x^{1} × x^{4} = -6x^{1 + 4} = **-6x ^{5}**

-2x × x = -2 × x × x = -2 × x

-2x × -6 = -2 × -6x = **12x**

5 × 3x^{4} = **15x ^{4}**

5 × x = **5x**

5 × -6 = **-30**

We get 12x^{7} + 4x^{4} + -24x^{3} + -6x^{5} + -2x^{2} + 12x + 15x^{4} + 5x + -30

Combine like terms

12x^{7} + (4x^{4} + 15x^{4}) + -24x^{3} + -6x^{5} + -2x^{2} + (12x + 5x) + -30

12x^{7} + 19x^{4} + -24x^{3} + -6x^{5} + -2x^{2} + 17x + -30

Write the polynomial in standard form

12x^{7} + -6x^{5} + 19x^{4} + -24x^{3} + -2x^{2} + 17x + -30

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

**Example #5:**

Multiply 3x + 4y by y + 2

(3x + 4y)(y + z) = 3x × y + 3x × z + 4y × y + 4y × z

(3x + 4y)(y + z) = 3xy + 3xz + 4y^{2} + 4yz

Notice that when the variables are different, just put them next to each other!