Multiplying polynomials

Multiplying polynomials with the following two good examples will help you master this topic once and for all. That is all you will need. It could be very useful though to review the multiplication of binomials

Example #1:

Multiply 4x³ + 2x + 5 by 3x⁴ + x + 6

(4x³ + 2x + 5) × (3x⁴ + x + 6)

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!

We will show the result of each multiplication or whatever will be added together in bold

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

Look at it carefully. Each term is separated by an addition sign.

So the first term is 4x³

The second term is 2x

The third term is 5

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

Again, each term is separated by an addition sign.

So 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.

Let's do it!

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

Example #2:

Multiply 4x³ − 2x + 5 by 3x⁴ + x − 6

Almost the same problem. 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 #1

(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

Pay careful attention to this when multiplying polynomials!

-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

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

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