1. Multiply a polynomial by a monomial.

2. Multiply a polynomial by a polynomial.

The Distributive PropertyLook at the following expression:

3(x + 7) This expression is the sum of x and 7 multiplied by 3.

To simplify this expression we can distribute the multiplication by 3 to each number in the sum.

(3 • x) + (3 • 7)

3x + 21

Whenever we multiply two numbers, we are putting the distributive property to work.

7(23) We can rewrite 23 as (20 + 3) then the problem would look like 7(20 + 3).

Using the distributive property:

(7 • 20) + (7 • 3) = 140 + 21 = 161

When we learn to multiply multi-digit numbers, we do the same thing in a vertical format.

23x____7

7 • 3 = 21. Keep the 1 in the ones position then carry the 2 into the tens position.

1

2

7 • 2 = 14. Add the 2 from before and we get 16.

16

What we’ve really done in the second step, is multiply 7 by 20, then add the 20 left over from the first step to get 160. We add this to the 1 to get 161.

Multiply: 3xy(2x + y)

This problem is just like the review problems except for a few more variables.

To multiply we need to distribute the 3xy over the addition.

3xy(2x + y) = (3xy • 2x) + (3xy • y) =

Then use the order of operations and the properties of exponents to simplify.

6x2y + 3xy2

We can also multiply a polynomial and a monomial using a vertical format in the same way we would multiply two numbers.

Multiply: 7x2(2xy – 3x2)

2xy – 3x2

7x2x________Align the terms vertically with the monomial under the polynomial.

Now multiply each term in the polynomial by the monomial.

– 21x414x3y

Keep track of negative signs.

To multiply a polynomial by another polynomial we use the distributive property as we did before.

Multiply: (x + 3)(x – 2)

Remember that we could use a vertical format when multiplying a polynomial by monomial. We can do the same here.

(x + 3)(x – 2)x________

Line up the terms by degree.

Multiply in the same way you would multiply two 2-digit numbers.

– 6 2x+ 0 + 3xx2_________– 6 + 5xx2

Multiply: (x + 3)(x – 2)

(x + 3)(x – 2)x________

– 6 2x+ 0 + 3xx2_________– 6 + 5xx2

To multiply the problem below, we have distributed each term in one of the polynomials to each term in the other polynomial.

Here is another example.(x2 – 3x + 2)(x2 – 3)

(x2 – 3x + 2)(x2 – 3)x____________

Line up like terms.

– 6 + 9x– 3x2

+ 0+ 0x+ 2x2– 3x3x4__________________– 6 + 9x – 1x2 – 3x3 x4

It is also advantageous to multiply polynomials without rewriting them in a vertical format.

Multiply: (x + 2)(x – 5)

Though the format does not change, we must still distribute each term of one polynomial to each term of the other polynomial.

Each term in (x+2) is distributed to each term in (x – 5).

(x + 2)(x – 5)

This pattern for multiplying polynomials is called FOIL.

Multiply the First terms.

Multiply the Outside terms.

Multiply the Inside terms.

Multiply the Last terms.

F

O

I

L After you multiply, collect like terms.

Example: (x – 6)(2x + 1)

x(2x) + x(1) – (6)2x – 6(1)

2x2 + x – 12x – 6

2x2 – 11x – 6

1. 2x2(3xy + 7x – 2y)

2. (x + 4)(x – 3)

3. (2y – 3x)(y – 2)

2x2(3xy + 7x – 2y)

2x2(3xy) + 2x2(7x) + 2x2(–2y)

2x2(3xy + 7x – 2y)

6x3y + 14x2 – 4x2y

(x + 4)(x – 3)

(x + 4)(x – 3)

x(x) + x(–3) + 4(x) + 4(–3)

x2 – 3x + 4x – 12

x2 + x – 12

(2y – 3x)(y – 2)

(2y – 3x)(y – 2)

2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)

2y2 – 4y – 3xy + 6x