D. Factoring Special Forms and Quadratic Forms
Tải bản đầy đủ - 0trang
cob19545_chR_040-053.qxd
7/28/10
2:38 PM
Page 44
College Algebra Graphs & Models—
44
R–44
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
cob19545_chR_040-053.qxd
7/28/10
2:38 PM
Page 45
College Algebra Graphs & Models—
R–45
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
ᮣ
Quadratic Forms and u-Substitution
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
ᮣ
Factoring a Quadratic Form
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
and quadratic 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.
cob19545_chR_040-053.qxd
11/22/10
10:56 AM
Page 46
College Algebra Graphs & Models—
46
R–46
CHAPTER R A Review of Basic Concepts and Skills
Factoring
Polynomials
Standard Form:
decreasing order of degree
Greatest Common Factor
(positive leading coefficient)
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)
Advanced
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.
cob19545_chR_040-053.qxd
7/28/10
2:38 PM
Page 47
College Algebra Graphs & Models—
R–47
47
Section R.4 Factoring Polynomials and Solving Polynomial Equations by Factoring
Quadratic Equations
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
Latin word quadratum, meaning
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
the leading term is squared.
ᮣ
Standard Form
Quadratic
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.