5.3 multiplying polynomials: special products · to multiply polynomials, you use many of the rules...

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5.3 Multiplying Polynomials: Special Products

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What You Will Learn

Find products with monomial multipliers

Multiplying binomials using the Distributive

Property and the FOIL Method

Multiply polynomials using a horizontal or

vertical format

Identify and use special binomial products

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Monomial Multipliers

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Monomial Multipliers

To multiply polynomials, you use many of the rules for

simplifying algebraic expressions.

1. The Distributive Property

2. Combining like terms

3. Removing symbols of grouping

4. Rules of exponents

The simplest type of polynomial multiplication involves a

monomial multiplier.

The product is obtained by direct application of the

Distributive Property.

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Monomial Multipliers

For instance, to multiply the monomial x by the polynomial

(2x + 5), multiply each term of the polynomial by x.

(x)(2x + 5) = (x)(2x) + (x)(5) = 2x2 + 5x

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Example 1 – Finding Products with Monomial Multipliers

Find each product.

a. (3x – 7)(–2x)

b. 3x2(5x – x3 + 2)

c. (–x)(2x2 – 3x)

Solution:

a. (3x – 7)(–2x) = 3x(–2x) – 7(–2x)

= –6x2 + 14x

Distributive Property

Write in standard form.

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cont’d

b. 3x2(5x – x3 + 2)

= (3x2)(5x) – (3x2)(x3) + (3x2)(2)

= 15x3 – 3x5 + 6x2

= –3x5 + 15x3 + 6x2

c. (–x)(2x2 – 3x) = (–x)(2x2) – (–x)(3x)

= –2x3 + 3x2

Distributive Property

Rules of exponents

Write in standard form.

Distributive Property

Write in standard form.

Example 1 – Finding Products with Monomial Multipliers

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Multiplying Binomials

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Multiplying Binomials

To multiply two binomials, you can use both (left and right)

forms of the Distributive Property.

For example, if you treat the binomial (5x + 7) as a single

quantity, you can multiply (3x – 2) by (5x + 7) as follows.

(3x – 2)(5x + 7) = 3x(5x + 7) – 2(5x + 7)

= (3x)(5x) + (3x)(7) – (2)(5x) – 2(7)

= 15x2 + 21x – 10x – 14

= 15x2 + 11x – 14

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Multiplying Binomials

With practice, you should be able to multiply two binomials

without writing out all of the steps above.

In fact, the four products in the boxes above suggest that

you can write the product of two binomials in just one step.

This is called the FOIL Method. Note that the words first,

outer, inner, and last refer to the positions of the terms in

the original product.

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Example 2 – Multiplying Binomials with the Distributive Property

Use the Distributive Property to find each product.

a. (x – 1)(x + 5)

b. (2x + 3) (x – 2)

Solution:

a. (x – 1)(x + 5) = x(x + 5) – 1(x + 5)

= x2 + 5x – x – 5

= x2 + (5x – x) – 5

= x2 + 4x – 5

Right Distributive Property

Left Distributive Property

Group like terms.

Combine like terms.

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cont’d

Example 2 – Multiplying Binomials with the Distributive Property

b. (2x + 3)(x – 2) = 2x(x – 2) + 3(x – 2)

= 2x2 – 4x + 3x – 6

= 2x2 + (–4x + 3x) – 6

= 2x2 – x – 6

Right Distributive Property

Left Distributive Property

Group like terms.

Combine like terms.

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Example 3 – Multiplying Binomials using the FOIL Method

Use the FOIL Method to find each product.

a. (x + 4)(x – 4)

b. (3x + 5)(2x + 1)

Solution:

F O I L

a. (x + 4)(x – 4) = x2 – 4x + 4x – 16

= x2 – 16

Note that the outer and inner products add up to zero.

Combine like terms.

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Example 3 – Multiplying Binomials using the FOIL Method

cont’d

F O I L

b. (3x + 5)(2x + 1) = 6x2 + 3x + 10x + 5

= 6x2 + 13x + 5

Combine like terms.

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Example 4 – A Geometric Model of a Polynomial Product

Use the geometric model to show that

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

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Example 4 – A Geometric Model of a Polynomial Product

cont’d

Solution

The left part of the model shows that the sum of the areas

of the six rectangle is

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

The right part of the model shows that the area of the

rectangle is

(x + 1)(x + 2) = x2 + 2x + x + 2

= x2 + 3x + 2

So, x2 + 3x + 2 = (x + 1)(x + 2)

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Example 5 – Simplifying a Polynomial Expression

Simplify the expression and write the result in standard form

(4x + 5)2

Solution

(4x + 5)2 = (4x + 5)(4x + 5) Repeated multiplication

= 16x2 + 20x + 20x + 25 Use FOIL Method

= 16x2 + 40x + 25 Combine like terms

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Example 6 – Simplifying a Polynomial Expression

Simplify the expression and write the result in standard form

(3x2 – 2)(4x + 7) – (4x)2

Solution

(3x2 – 2)(4x + 7) – (4x)2 = 12x3 + 21x2 – 8x – 14 – (4x)2 Use FOIL Method

= 12x3 + 21x2 – 8x – 14 – 16x2 Square monomial

= 12x3 + 5x2 – 8x – 14 Combine like terms

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Multiplying Polynomials

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Multiplying Polynomials

The FOIL Method for multiplying two binomials is simply a

device for guaranteeing that each term of one binomial is

multiplied by each term of the other binomial.

(ax + b)(cx + d) = ax(cx) + ax(d) + b(cx) + b(d)

F O I L

This same rule applies to the product of any two

polynomials: each term of one polynomial must be

multiplied by each term of the other polynomial.

This can be accomplished using either a horizontal or a

vertical format.

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Example 7 – Multiplying Polynomials Horizontally

Use a horizontal format to find each product.

a. (x – 4)(x2 – 4x + 2)

b. (2x2 – 7x + 1)(4x + 3)

Solution:

a. (x – 4)(x2 – 4x + 2)

= x(x2 – 4x + 2) – 4(x2 – 4x + 2)

= x3 – 4x2 + 2x – 4x2 + 16x – 8

= x3 – 8x2 + 18x – 8

Combine like terms.

Distributive Property

Distributive Property

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Example 7 – Multiplying Polynomials Horizontally

cont’d

b. (2x2 – 7x + 1)(4x + 3)

= (2x2 – 7x + 1)(4x) + (2x2 – 7x + 1)(3)

= 8x3 – 28x2 + 4x + 6x2 – 21x + 3

= 8x3 – 22x2 – 17x + 3

Combine like terms.

Distributive Property

Distributive Property

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Example 10 – Raising a Polynomial to a Power

Use two steps to expand (x – 3)3

Solution:

Step 1: (x – 3)2 = (x – 3)(x – 3)

= x2 – 3x – 3x + 9

= x2 – 6x + 9

Step 2: (x2 – 6x + 9)(x – 3) = (x2 – 6x + 9)(x) – (x2 – 6x + 9)(3)

= x3 – 6x2 + 9x – 3x2 + 18x – 27

= x3 – 9x2 + 27x – 27

So, (x – 3)3 = x3 – 9x2 + 27x – 27

Combine like terms

Use FOIL Method

Repeated multiplication

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Special Products

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Special Products

Some binomial products, such as those in Example 3(a),

has special forms that occur frequently in algebra.

The product

(x + 4)(x – 4)

is called a product of the sum and difference of two

terms.

With such products, the two middle terms cancel, as

follows.

(x + 4)(x – 4) = x2 – 4x + 4x – 16

= x2 – 16

Sum and difference of two terms

Product has no middle term.

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Special Products

Another common type of product is the square of a

binomial.

(4x + 5)2 = (4x + 5)(4x + 5)

= 16x2 + 20x + 20x + 25

= 16x2 + 40x + 25

Square of a binomial

Use FOIL Method.

Middle term is twice the product

of the terms of the binomial.

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Special Products

In general, when a binomial is squared, the resulting middle

term is always twice the product of the two terms.

(a + b)2 = a2 + 2(ab) + b2

Be sure to include the middle term.

For instance, (a + b)2 is not equal to a2 + b2.

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Special Products

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Example 11 – Finding Special Products

a. (5x – 6)(5x + 6) = (5x)2 – (6) 2 = 25x2 – 36

b. (3x + 7)2 = (3x)2 + 2(3x)(7) + (7)2 = 9x2 + 42x + 14

c. (4x + 9)2 = (4x)2 + 2(4x)(9) + (9)2 = 16x2 + 72x + 81

d. (6 + 5x2)2 = (4)2 – 2(6)(5x2) + (5x2)2

= 36 – 60x2 + (5) 2(x2) 2 = 36 – 60x2 + 25x4

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Example 12 – Finding the Dimensions of a Golf Tee

A landscaper wants to reshape a square tee area for the

ninth hole of a golf course. The new tee area will have one

side 2 feet longer and the adjacent side 6 feet longer than

the original tee. The area of the new tee will be 204 square

feet greater than the area of the original tee. What are the

dimensions of the original tee?

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Solution

Verbal Model:

Labels: Original length = original width = x (feet)

Original area = x2 (square feet)

New length = x + 6 (feet)

New width = x + 2 (feet)

Equation: (x + 6)(x + 2) = x2 + 204 Write equation

x2 + 8x + 12 = x2 + 204 Multiply factors

8x + 12 = 204 Subtract x2 from each side

8x = 192 Subtract 12 from each side

x = 24 Divide each side by 8

cont’d

Example 12 – Finding the Dimensions of a Golf Tee

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Page 244 #’s 1 – 10 down the column

Page 245 #’s 19 & 23

Page 247 #’s 31 – 39 down the column

Page 248 #’s 43 – 47 down the column

Page 249 #’s 55 & 59

Page 251 #’s 77 & 78

Homework: