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D. Factoring Special Forms and Quadratic Forms

# D. Factoring Special Forms and Quadratic Forms

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CHAPTER R A Review of Basic Concepts and Skills

Factoring Perfect Square Trinomials

Given any expression that can be written in the form A2 Ϯ 2AB ϩ B2,

1. A2 ϩ 2AB ϩ B2 ϭ 1A ϩ B2 2

2. A2 Ϫ 2AB ϩ B2 ϭ 1A Ϫ B2 2

EXAMPLE 7

Factoring a Perfect Square Trinomial

Factor 12m3 Ϫ 12m2 ϩ 3m.

Solution

12m3 Ϫ 12m2 ϩ 3m

ϭ 3m14m2 Ϫ 4m ϩ 12

check for common factors: GCF ϭ 3m

factor out 3m

For the remaining trinomial 4m2 Ϫ 4m ϩ 1 p

1. Are the first and last terms perfect squares?

4m2 ϭ 12m2 2 and 1 ϭ 112 2 ✓ Yes.

2. Is the linear term twice the product of 2m and 1?

2 # 2m # 1 ϭ 4m ✓ Yes.

Factor as a binomial square: 4m2 Ϫ 4m ϩ 1 ϭ 12m Ϫ 12 2

This shows 12m3 Ϫ 12m2 ϩ 3m ϭ 3m12m Ϫ 12 2.

Now try Exercises 19 and 20

CAUTION

As shown in Example 7, be sure to include the GCF in your final answer. It is a common

error to “leave the GCF behind.”

In actual practice, these calculations can be performed mentally, making the

process much more efficient.

Sum or Difference of Two Perfect Cubes

Recall that the difference of two perfect squares is factorable, but the sum of two perfect

squares is prime. In contrast, both the sum and difference of two perfect cubes are factorable. For either A3 ϩ B3 or A3 Ϫ B3 we have the following:

1. Each will factor into the product of a binomial

and a trinomial:

2. The terms of the binomial are the quantities

being cubed:

3. The terms of the trinomial are the square of A,

the product AB, and the square of B, respectively:

4. The binomial takes the same sign as the original

expression

5. The middle term of the trinomial takes the

opposite sign of the original expression

(the last term is always positive):

(

)(

binomial

)

trinomial

(A

B)(

)

(A

B)(A2

AB

B 2)

(A Ϯ B)(A2

AB

B 2)

(A Ϯ B)(A2 ϯ AB ϩ B 2)

Factoring the Sum or Difference of Two Perfect Cubes: A3 Ϯ B3

1. A3 ϩ B3 ϭ 1A ϩ B2 1A2 Ϫ AB ϩ B2 2

2. A3 Ϫ B3 ϭ 1A Ϫ B21A2 ϩ AB ϩ B2 2

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Section R.4 Factoring Polynomials and Solving Polynomial Equations by Factoring

EXAMPLE 8

Factoring the Sum and Difference of Two Perfect Cubes

Factor completely:

a. x3 ϩ 125

Solution

45

a.

b.

b. Ϫ5m3n ϩ 40n4

x3 ϩ 125 ϭ x3 ϩ 53

Use A3 ϩ B3 ϭ 1A ϩ B21A2 Ϫ AB ϩ B2 2

x3 ϩ 53 ϭ 1x ϩ 52 1x2 Ϫ 5x ϩ 252

write terms as perfect cubes

factoring template

A S x and B S 5

Ϫ5m3n ϩ 40n4 ϭ Ϫ5n1m3 Ϫ 8n3 2

ϭ Ϫ5n 3m3 Ϫ 12n2 3 4

Use A3 Ϫ B3 ϭ 1A Ϫ B21A2 ϩ AB ϩ B2 2

m3 Ϫ 12n2 3 ϭ 1m Ϫ 2n2 3 m2 ϩ m12n2 ϩ 12n2 2 4

ϭ 1m Ϫ 2n21m2 ϩ 2mn ϩ 4n2 2

3

4

1 Ϫ5m n ϩ 40n ϭ Ϫ5n1m Ϫ 2n21m2 ϩ 2mn ϩ 4n2 2.

check for common

factors 1GCF ϭ Ϫ5n2

write terms as perfect cubes

factoring template

A S m and B S 2n

simplify

factored form

The results for parts (a) and (b) can be checked using multiplication.

Now try Exercises 21 and 22

For any quadratic expression ax2 ϩ bx ϩ c in standard form, the degree of the

leading term is twice the degree of the middle term. Generally, a trinomial is in

quadratic form if it can be written as a1 __ 2 2 ϩ b1 __ 2 ϩ c, where the parentheses

“hold” the same factors. The equation x4 Ϫ 13x2 ϩ 36 ϭ 0 is in quadratic form

since 1x2 2 2 Ϫ 131x2 2 ϩ 36 ϭ 0. In many cases, we can factor these expressions

using a placeholder substitution that transforms them into a more recognizable

form. In a study of algebra, the letter “u” often plays this role. If we let u represent

x2, the expression 1x2 2 2 Ϫ 131x2 2 ϩ 36 becomes u2 Ϫ 13u ϩ 36, which can be

factored into 1u Ϫ 92 1u Ϫ 42. After “unsubstituting” (replace u with x2), we have

1x2 Ϫ 92 1x2 Ϫ 42 ϭ 1x ϩ 321x Ϫ 321x ϩ 22 1x Ϫ 22.

EXAMPLE 9

Solution

Expanding the binomials would produce a fourth-degree polynomial that would

be very difficult to factor. Instead we note the expression is in quadratic

form. Letting u represent x2 Ϫ 2x (the variable part of the “middle” term),

1x2 Ϫ 2x2 2 Ϫ 21x2 Ϫ 2x2 Ϫ 3 becomes u2 Ϫ 2u Ϫ 3.

Write in completely factored form: 1x2 Ϫ 2x2 2 Ϫ 21x2 Ϫ 2x2 Ϫ 3.

u2 Ϫ 2u Ϫ 3 ϭ 1u Ϫ 32 1u ϩ 12

factor

To finish up, write the expression in terms of x, substituting x2 Ϫ 2x for u.

ϭ 1x2 Ϫ 2x Ϫ 32 1x2 Ϫ 2x ϩ 12

substitute x2 Ϫ 2x for u

The resulting trinomials can be further factored.

ϭ 1x Ϫ 32 1x ϩ 12 1x Ϫ 12 2

x2 Ϫ 2x ϩ 1 ϭ 1x Ϫ 12 2

Now try Exercises 23 and 24

D. You’ve just seen how

we can factor special forms

It is well known that information is retained longer and used more effectively

when it’s placed in an organized form. The “factoring flowchart” provided in Figure R.4 offers a streamlined and systematic approach to factoring and the concepts involved. However, with some practice the process tends to “flow” more naturally than

following a chart, with many of the decisions becoming automatic.

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CHAPTER R A Review of Basic Concepts and Skills

Factoring

Polynomials

Standard Form:

decreasing order of degree

Greatest Common Factor

Number of Terms

Two

Difference

of squares

Difference

of cubes

Four

Three

Sum

of cubes

Trinomials

(a ϭ 1)

• Can any result be factored further?

Trinomials

(a 1)

methods

(Section 4.2)

Factor by

grouping

• Polynomials that cannot be factored are said to be prime.

Figure R.4

For additional practice with these ideas, see Exercises 25 through 52.

E. Polynomial Equations and the Zero Product Property

The ability to solve linear and quadratic equations is the foundation on which a large

percentage of our future studies are built. Both are closely linked to the solution of

other equation types, as well as to the graphs of these equations.

In standard form, linear and quadratic equations have a known number of

terms, so we commonly represent their coefficients using the early letters of the

alphabet, as in ax2 ϩ bx ϩ c ϭ 0. However, these equations belong to the larger

family of polynomial equations. To write a general polynomial, where the number of terms is unknown, we often represent the coefficients using subscripts on a

single variable, such as a1, a2, a3, and so on.

Polynomial Equations

A polynomial equation of degree n is one of the form

anxn ϩ anϪ1xnϪ1 ϩ p ϩ a1x1 ϩ a0 ϭ 0

where an, anϪ1, p , a1, a0 are real numbers and an

0.

As a prelude to solving polynomial equations of higher degree, we’ll first look at

quadratic equations and the zero product property. As before, a quadratic equation is

one that can be written in the form ax2 ϩ bx ϩ c ϭ 0, where a, b, and c are real numbers and a 0. As written, the equation is in standard form, meaning the terms are

in decreasing order of degree and the equation is set equal to zero.

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Section R.4 Factoring Polynomials and Solving Polynomial Equations by Factoring

A quadratic equation can be written in the form

ax2 ϩ bx ϩ c ϭ 0,

with a, b, c ʦ ‫ޒ‬, and a

0.

Notice that a is the leading coefficient, b is the coefficient of the linear (first

degree) term, and c is a constant. All quadratic equations have degree two, but can have

one, two, or three terms. The equation n2 Ϫ 81 ϭ 0 is a quadratic equation with two

terms, where a ϭ 1, b ϭ 0, and c ϭ Ϫ81.

EXAMPLE 10

Determining Whether an Equation Is Quadratic

State whether the given equation is quadratic. If yes, identify coefficients a, b, and c.

Ϫ3

a. 2x2 Ϫ 18 ϭ 0

b. z Ϫ 12 Ϫ 3z2 ϭ 0

c.

xϩ5ϭ0

4

d. z3 Ϫ 2z2 ϩ 7z ϭ 8

e. 0.8x2 ϭ 0

Solution

WORTHY OF NOTE

The word quadratic comes from the

square. The word historically refers

to the “four sidedness” of a square,

but mathematically to the area of a

square. Hence its application to

polynomials of the form

ax2 ϩ bx ϩ c, where the variable of

Standard Form

a.

2x Ϫ 18 ϭ 0

yes, deg 2

aϭ2

b.

Ϫ3z ϩ z Ϫ 12 ϭ 0

yes, deg 2

a ϭ Ϫ3

c.

Ϫ3

xϩ5ϭ0

4

no, deg 1

(linear equation)

d.

z3 Ϫ 2z2 ϩ 7z Ϫ 8 ϭ 0

no, deg 3

(cubic equation)

e.

0.8x ϭ 0

yes, deg 2

2

2

2

Coefficients

bϭ0

a ϭ 0.8

c ϭ Ϫ18

bϭ1

bϭ0

c ϭ Ϫ12

cϭ0

Now try Exercises 53 through 64

With quadratic and other polynomial equations, we generally cannot isolate the

variable on one side using only properties of equality, because the variable is raised to

different powers. Instead we attempt to solve the equation by factoring and applying

the zero product property.

Zero Product Property

If A and B represent real numbers or real-valued expressions

and A # B ϭ 0,

then A ϭ 0 or B ϭ 0.

In words, the property says, If the product of any two (or more) factors is equal to

zero, then at least one of the factors must be equal to zero. We can use this property to

solve higher degree equations after rewriting them in terms of equations with lesser

degree. As with linear equations, values that make the original equation true are called

solutions or roots of the equation.

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D. Factoring Special Forms and Quadratic Forms

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