Multiplication of Polynomials

Using what we already know!

REVIEW

What property would be used to simplify the following expression:

)62(4 x

248 x6424 x

The Distributive Property

Connecting

Recall our definition of a monomial:

– A number, variable, or the product of a number and one or more variables

How did this differ from a polynomial?

– A polynomial adds (or subtracts) 2 or more monomials.

By using the distributive property, we have actually multiplied a monomial by a polynomial.

An Alternative

Suppose I need to simplify:

)53(2 xx

xx 106 2

While distributive property will work, another option is to use generic rectangles!

2x

3x 5

Once we find the area of the smaller rectangles, we can add them to get

the area of the entire large rectangle.

6x2 10x

The factors are the base and height of the rectangle.

Remember Area = base • height

PRACTICE

Simplify each of the following:

)52(3 yy

)73(4 aa

)62(5 2 yxxy

Once you have the solutions for

each expression, click your

mouse again to see the

solutions.

xyxyyx

aa

yy

30105

2812

156

23

2

2

Taking the next step

What happens if we want to multiply two binomials?

– What makes a binomial different from a monomial?

A binomial adds 2 monomials together

Ex. 2x + 5

Consider the following:

)2)(5( xx

When multiplying binomials, we have 2 different methods to choose from:

Generic Rectangles

FOIL

Generic Rectangles

Because we have 4 terms, we need to break the rectangle into 4 sections.

)2)(5( xx

Each factor represents the base and the height of the rectangle.

x

+

5

x + 2

Find the area of the smaller rectangles.

x2 2x

5x 10

Add the areas together to get the total area of the rectangle.

1071052 22 xxxxx

FOIL

F: x • x = x2

O: x • 2 = 2x I: 5 • x = 5x L: 5 • 2 = 10

107

10522

2

xx

xxx

The letters in FOIL represents the position of the terms in the expression:

)2)(5( xx

First: The x terms are 1st in each factor.Outside: The x and 2 are on the outside of the expression.

Inside: The 5 and x are on the inside of the expression.

Last: The 5 and 2 are the last terms in each factor.

Once you identify the terms, multiply them.

Add your solutions together and simplify.

F F

O

OI

IL L

Watch out for the signs

How do your answers change when the signs change?

(x – 5) (x – 2)

(x + 5) (x – 2)

(x – 5) (x + 2)

1072 xx

1032 xx

1032 xx

Try each of the problems using the method of your choice. Click your

mouse to get the solutions.

Find the pattern

What is the pattern with with the signs?

– When both are positive, the answer has 2 addition signs.

(x+5)(x+2) = x2+7x+10– When both are negative, the

second sign is negative, the third is positive

(x - 5)(x - 2) = x2 - 7x+10– When the signs are different,

the third sign is negative, the second sign depends on the terms.

(x - 5)(x+2) = x2 - 3x – 10 (x+5)(x - 2) = x2+3x - 10

Another example

Consider the following:– (2x – 3)(2x + 3)– What do you notice about

this problem?– These two factors are

known as CONJUGATES Same terms separated

by different signs

What happens when you multiply two conjugates?

– The middle term gets eliminated!!

– (2x – 3)(2x + 3)

94 2 x

9664 2 xxx

Practice

Multiply the following using your method of choice. When you are finished, Click the mouse again to see the solutions:

– (2y + 4)(y – 3)

– (-3m+6)(2m+1)

– (5n – 2)(n – 7)

– (w – 5)(2w + 1)592

14375

696

1222

2

2

2

2

ww

nn

mm

yy