Multiplying Polynomials -Distributive property -FOIL -Box Method

Multiplying Polynomials -Distributive property -FOIL -Box Method

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To multiply a polynomial by a monomial, use the distributive

property and the rule for multiplying exponential expressions.

Examples: 1. Multiply: 2x(3x2 + 2x – 1).

= 6x3 + 4x2 – 2x

2. Multiply: – 3x2y(5x2 – 2xy + 7y2).

= – 3x2y(5x2 ) – 3x2y(– 2xy) – 3x2y(7y2)

= – 15x4y + 6x3y2 – 21x2y3

= 2x(3x2 ) + 2x(2x) + 2x(–1)

3

Multiply: – 2xy(8x2 +2xy – 5y2).

= – 2xy(8x2 ) – 2xy( 2xy) – 2xy(-5y2)

= – 16x3y – 6x2y2 + 10xy3

Multiplying Polynomials-Try it!

Try it!.

Multiply the polynomial by the monomial.

1) 3(x + 4)

2)

3) 26k(2k 4k 3)

2a(a 5)3x 12

22a 10a

3 212k 24k 18k

Distributive Property

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To multiply two binomials use a method called FOIL,

which is based on the distributive property. The letters

of FOIL stand for First, Outer, Inner, and Last.

1. Multiply the first terms.

3. Multiply the inner terms.

4. Multiply the last terms.

2. Multiply the outer terms.

5. Combine like terms.

For use with the product of binomials only!

(x 3)(x 5)

First Outer Inner Last

2x

For use with the product of binomials only!

(x 3)(x 5)

First Outer Inner Last

2x 5x

For use with the product of binomials only!

(x 3)(x 5)

First Outer Inner Last

2x 5x 3x

For use with the product of binomials only!

(x 3)(x 5)

First Outer Inner Last2x 5x 3x 15

For use with the product of binomials only!

(x 3)(x 5)

First Outer Inner Last

2x 5x 3x 15

2x 2x 15

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Examples: 1. Multiply: (2x + 1)(7x – 5).

= 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5)

= 14x2 – 10x + 7x – 5

2. Multiply: (5x – 3y)(7x + 6y).

= 35x2 + 30xy – 21yx – 18y2

= 14x2 – 3x – 5

= 35x2 + 9xy – 18y2

= 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y)

First Outer Inner Last

First Outer Inner Last

Try...

(m 3)(m 6)

First Outer Inner Last

2m 6m 3m 18

2m 9m 18

The third method is the Box Method. This method works for every problem!

Here’s how you do it. Multiply (3x – 5)(5x + 2)

Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where.

This will be modeled in the next problem along with

FOIL.

3x -5

5x

+2

Multiply (3x - 5)(5x + 2)

First terms:Outer terms:Inner terms:Last terms: Combine like terms.

15x2 - 19x – 10

3x -5

5x

+2

15x2

+6x

-25x

-10

You have 3 techniques. Pick the one you like the best!

15x2

+6x-25x-10

Try it! Multiply (7p - 2)(3p - 4)

First terms:Outer terms:Inner terms:Last terms: Combine like terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p-6p+8

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To multiply two polynomials, apply the distributive property.

Example: Multiply: (x – 1)(2x2 + 7x + 3).

= (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3)

= 2x3 – 2x2 + 7x2 – 7x + 3x – 3

= 2x3 + 5x2 – 4x – 3

Two polynomials can also be multiplied using a vertical format.

– 2x2 – 7x – 3 2x3 + 7x2 + 3x

2x3 + 5x2 – 4x – 3x

Multiply – 1(2x2 + 7x + 3).

Multiply x(2x2 + 7x + 3).

Add the terms in each column.

2x2 + 7x + 3x – 1

Example:

Multiply (2x - 5)(x2 - 5x + 4)You cannot use FOIL because they are not

BOTH binomials. You must use the distributive property.

2x(x2 - 5x + 4) - 5(x2 - 5x + 4)2x3 - 10x2 + 8x - 5x2 + 25x - 20

Group and combine like terms.2x3 - 10x2 - 5x2 + 8x + 25x - 20

2x3 - 15x2 + 33x - 20

x2 -5x +4

2x

-5

Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH

binomials. You must use the distributive property or box method.

2x3

-5x2

-10x2

+25x

+8x

-20

Almost

done!

Go to

the next

slide!

x2 -5x +4

2x

-5

Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!

2x3

-5x2

-10x2

+25x

+8x

-20

2x3 – 15x2 + 33x - 20

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Example: The length of a rectangle is (x + 5) ft. The width

is (x – 6) ft. Find the area of the rectangle in terms of

the variable x.

A = L · W = Area

x – 6

x + 5

L = (x + 5) ft

W = (x – 6) ft

A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30

= x2 – x – 30

The area is (x2 – x – 30) ft2.