6.4 factoring polynomials with special forms · 1/6/2017 · the rule a2 2– b = (a + b)(a – b)...

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6.4 Factoring Polynomials with Special Forms

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What You Will Learn

Factor the difference of two squares

Factor a polynomial completely

Identify and factor perfect square trinomials

Factor the sum or difference of two cubes

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Difference of Two Squares

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One of the easiest special polynomial forms to recognize

and to factor is the form a2 – b2.

It is called a difference of two squares and it factors

according to the following pattern.

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To recognize perfect square terms, look for coefficients that

are squares of integers and for variables raised to even

powers.

Here are some examples.

Original Difference

Polynomial of Squares Factored Form

x2 – 1 (x)2 – (1)2 (x + 1)(x – 1)

4x2 – 9 (2x)2 – (3)2 (2x + 3)(2x – 3)

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Example 1 – Factoring the Difference of Two Square

Factor each polynomial.

a. x2 – 36

b. x2 –

c. 81x2 – 49

Solution:

a. x2 – 36 = x2 – 62

= (x + 6)(x – 6)

Write as difference of two squares.

Factored form

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cont’d

b. x2 – = x2 –

=

c. 81x2 – 49 = (9x)2 – 72

= (9x + 7)(9x – 7)

Check your results by using the FOIL Method.


Factored form

Factored form


Example 1 – Factoring the Difference of Two Square

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The rule a2 – b2 = (a + b)(a – b) applies to polynomials or

expressions in which a and b are themselves expressions.

Sometimes the difference of two squares can be hidden by

the presence of a common monomial factor.

Remember that with all factoring techniques, you should

first remove any common monomial factors.


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Example 2 – Removing a Common Monomial Factor First

A hammer is dropped from the roof of a building. The

height of the hammer is given by the expression –16t2 + 64,

where t is the time in seconds.

a. Factor the expression

b. How many seconds does it take the hammer to fall to a

height of 41 feet?

Solution

a. –16t2 + 64 = –16(t2 – 4)

= –16(t2 – 22)

= –16(t + 2)(t – 2)

Factor out common monomial factor

Write a difference of two squares

Factored form

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b. Use a spreadsheet to find

the heights of the hammer

at 0.1-second intervals of

the time t.

From the spreadsheet, you

can see that the hammer

falls to a height of 41 feet in

about 1.2 seconds.

cont’d

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Factoring Completely

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To factor a polynomial completely, you should always

check to see whether the factors obtained might

themselves be factorable.

That is, can any of the factors be factored? For instance,

after factoring the polynomial (x4 – 1) once as the

difference of two squares

(x4 – 1) = (x2)2 – 12

= (x2 + 1)(x2 – 1)

you can see that the second factor is itself the difference of

two squares.

Factored form


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So, to factor the polynomial completely, you must continue

the factoring process.

(x4 – 1) = (x2 + 1)(x2 – 1)

= (x2 + 1)(x + 1)(x – 1)

Factor completely.

Factor as difference of two squares.

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Factor the polynomial 20x3 – 5x.

Solution:

20x3 – 5x = 5x(4x2 – 1)

= 5x[(2x)2 – 12]

= 5x(2x + 1)(2x – 1)

Factor out common monomials

factor 5x Write as difference of two

squares

Factored form

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Example 4 – Factoring Completely

Factor the polynomial x4 – 16 completely.

Solution:

Recognizing x4 – 16 as a difference of two squares, you can write

x4 – 16 = (x2)2 – 42

= (x2 + 4)(x2 – 4).

Note that the second factor (x2 – 4) is itself a difference of

two squares, and so

x4 – 16 = (x2 + 4)(x2 – 4)

= (x2 + 4)(x + 2)(x – 2).


Factored form

Factor as difference of two squares.

Factor completely.

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Perfect Square Trinomials

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Perfect Square Trinomial

A perfect square trinomial is the square of a binomial.

For instance,

x2 + 4x + 4 = (x + 2)(x + 2)

= (x + 2)2

is the square of the binomial (x + 2).

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Perfect square trinomials come in two forms: one in which

the middle term is positive and the other in which the

middle term is negative.

In both cases, the first and last terms are positive perfect

squares.


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Example 5 – Identifying Perfect Square Trinomials

Which of the following are perfect square trinomials?

a. m2 – 4m + 4

b. 4x2 – 2x + 1

c. y2 + 6y – 9

d. x2 + x +

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cont’d

Example 5 – Identifying Perfect Square Trinomials

a. This polynomial is a perfect square trinomial.

It factors as (m – 2)2.

b. This polynomial is not a perfect square trinomial

because the middle term is not twice the product of

2x and 1.

c. This polynomial is not a perfect square trinomial because

the last term, –9, is not positive.

d. This polynomial is a perfect square trinomial.

It factors as .

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Example 6 – Factoring a Perfect Square Trinomial

Factor the trinomial y2 – 6y + 9.

Solution

y2 – 6y + 9 = y2 – 2(3y) + 9 Recognize the pattern

= (y – 3) 2 Write in factored form

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Factor the trinomial 16x2 + 40y + 25.

Solution

16x2 + 40y + 25 = (4x)2 + 2(4x)(5) + 52 Recognize the pattern

= (4x + 5)2 Write in factored form

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Factor the trinomial 9x2 + 24xy + 16y2.

Solution

9x2 + 24xy + 16y2 = (3x)2 + 2(3x)(4y) + (4y)2

= (3x + 4y)2

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Sum or Difference of Two Cubes

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The last type of special factoring presented in this section

is the sum or difference of two cubes.

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Example 9 – Factoring the Sum or Difference of two Cubes, a.

Factor the polynomial y3 + 27.

Solution:

y3 + 27 = y3 + 33

= (y + 3) [y2 – (y)(3) + 32]

= (y + 3)(y2 – 3y + 9)

Write as sum of two cubes.

Factored form

Simplify.

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Example 9 – Factoring the Sum or Difference of two Cubes, b.

Factor the polynomial 64 – x3.

Solution:

64 – x3 = y3 + 33

= (4 – x)[42 + (4)(x) + x2]

= (4 – x)(16 + 4x + x2)


Factored form

Simplify.

cont’d

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Example 9 – Factoring the Sum or Difference of two Cubes, c.

Factor the polynomial 2x3 – 16.

Solution:

2x3 – 16 = 2(x3 – 8)

= 2(x3 – 23)

= 2(x – 2)[x2 + (x)(2) + 22]

= 2(x – 2)(x2 + 2x + 4)

Factor out common monomial

factor 2


Factored form

Simplify

cont’d

Homework:

6 # 1, 2, 5, 6

Page 298 # 11 – 20 down

Page 300 # 31 – 40 down

Page 301 # 45 – 53 down

6.4 factoring polynomials with special forms · 1/6/2017 · the rule a2 2– b = (a + b)(a – b)...

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