Download Algebra Notes - Rochester Community Schools

Algebra Notes
Objective(s):
Section 9.1: Add and Subtract Polynomials
To be able to add and subtract polynomials.
Recall:
Coefficient (p. 97):
The number part of a term with a variable part.
Term of a polynomial (p. 97):
The parts of an expression that are added together.
Like Terms (p. 97):
Terms that have the same variable parts. Constant terms are also
like terms.
Vocabulary :
I. Monomial
A
number
variable
,
whole number
one or more variables with
II. Degree of a Monomial:
The
sum
, or the product of a number and
exponents
of the
exponents.
variables
of the
in the monomial.
III. Polynomial:
A
a
IV. Degree of a Polynomial: The
V. Leading Coefficient:
monomial
or a
term
sum
of monomials, each called
of the polynomial.
greatest degree of its terms
.
When a polynomial is written so that the exponents of a variable decrease from left to
right, the leading coefficient is the
coefficient of the first term
.
Rewriting a polynomial so that the exponents of a variable decrease from left
to right is often referred to as writing a polynomial in descending order of
exponents.
Example:
2x3 + x2 − 5x + 12
4
This polynomial has
2
The leading coefficient is
The degree is
3
The constant term is
VI. Binomial:
A polynomial with
2
terms.
VII. Trinomial:
A polynomial with
3
terms.
VIII. Adding Polynomials:
To add polynomials,
IX. Subtracting Polynomials: To subtract polynomials,
add like terms
terms.
.
.
12 .
.
add its opposite (multiply each term by −1) .
Notes 9.1
Examples:
1. Consider the polynomial 3x3 − 4x4 + x2.
a. What is the degree of the polynomial?
4
b. How many terms does this polynomial have?
3
c. Classify the polynomial according to the number of terms.
trinomial
d. Rewrite the polynomial in descending order of exponents.
−4x4 + 3x3 + x2
e. What is the leading coefficient of the polynomial?
f. List all of the coefficients of this polynomial.
g. List the terms of the polynomial.
−4
−4, 3, 1
−4x4, 3x3 , x2
2. Tell whether the expression is a polynomial. If it is, find its degree and classify it by the number of its terms.
Otherwise, tell why it is not a polynomial.
Expression
Is it a polynomial?
Classify by degree and
number of terms
4x
Yes
1st degree monomial
2x + 3x5 + 1
Yes
5th degree trinomial
1
7m 2
+m
x−4 + 3
8xy + 3x2y
No.
Exponent must be a
whole number
No. Exponent cannot be negative.
Yes
3rd degree binomial
Notes 9.1 page 3
3. Find the sum or difference.
a. (−2x2 + 3x − x3) + (3x2 + x3 − 12)
x2 + 3x − 12
b. (4x3 + 2x2 − 4) + (x3 − 3x2 + x)
5x3 − x2 + x − 4
c. (2m2 − 8) − (3m2 − 4m + 1)
−m2 + 4m − 9
d. (5y2 + 2y − 4) − (−y2 + 4y − 3)
6y2 − 2y − 1
4. During the period 1999 – 2005, the number of hours an individual person watched broadcast television B and cable
and satellite television C can be modeled by B = 2.8t2 − 35t + 879 and C = −5t2 + 80t + 712, where t is the
number of years since 1999.
a. Write a polynomial that represents the total number of hours of broadcast and cable watched.
B + C = −2.2t2 + 45t + 1591
b. About how many hours did people watch in 2002?
2002 is 3 years since 1999, so t = 3
If t = 3, then B + C = − 2.2(3)2 + 45(3) + 1591
= 1706.2 hours
Algebra Notes
Objective(s):
Section 9.2: Multiply Polynomials
To be able to multiply polynomials.
Vocabulary :
I. Recall properties of multiplying and adding expressions:
Examples:
2x • 4x =
8x2
2x + 4x =
6x
2x • 3x2 =
6x3
2x + 3x2 =
3x2 + 2x
2x2y3 + 4y3x2 =
6x2y3
2x3y2 + 4y3x2 =
2x3y2 + 4x2y3
2x(4x + 1) =
8x2 + 2x
II. FOIL Pattern:
F
O
(2x + 3)(4x + 1) =
F
O
I
L
2
8x + 2x + 12x + 3 = 8x2 + 14x + 3
I
L
Examples:
1. Find the product.
a.
3x2(2x3 − x2 + 4x − 3)
6x5 − 3x4 + 12x3 − 9x2
c. (2x − 1)(3x −4)
6x2 − 11x + 4
e. (x2 − x − 2)(3x − 1)
3x3 − 4x2 − 5x + 2
b. (x + 4)(2x − 1)
2x2 + 7x − 4
d. (4x + 3)(x + 2)
4x2 + 11x + 6
Notes 9.2
2. Perform the indicated operation.
a. (2x + 1) + (3x − 2)
b. (2x + 1)(3x − 2)
5x − 1
6x2 − x − 2
3. A rectangle has dimensions x + 3 and x + 5. Which expression shows the area of the rectangle?
A. x2 + 15
B. x2 + 3x + 15
C. x2 + 8x + 1
D. x2 + 8x
E. None of these
A = length x width
=(x + 3)(x + 5)
= x2 + 8x + 15
4. A rectangular trivet has a ceramic center and a wooden border.
The dimensions of the center and border are shown in the diagram.
x inches
a. Write a polynomial that represents
the total area of the trivet.
A = length x width
= (2x + 8) (2x + 6)
= 4x2 + 28x + 48
8”
x inches
6”
b. What is the total area of the trivet if
the width of the border is 2 inches?
A = 4(2)2 + 28(2) + 48
= 120 in2
5. Write a polynomial that represents the area of the shaded region.
A = (2x - 1) (x + 2) ‒ 10  8
= 2x2 + 4x ‒ x ‒ 2 ‒ 80
= 2x2 + 3x ‒ 82
8
10
2x - 1
x+2
Algebra Notes
Section 9.3: Finding Special Products of Polynomials
To use special product patterns to multiply polynomials.
Objective(s):
Vocabulary :
I. Square of a Binomial Pattern:
(write this on your formula sheet)
II. Sum and Difference Pattern:
(write this on your formula sheet)
(a + b)2 =
a2 + 2ab + b2
(a − b)2 =
a2 − 2ab + b2
a2 − b2
(a + b)(a − b) =
Examples:
1. Find the product.
a. (2x + 5)2
4x2
b. (3x − y)2
+ 20x + 25
d. (4x + y)(4x − y)
16x2
−
9x2
− 6xy +
e. (x + 1)(x + 1)
y2
x2
+ 2x + 1
2. Which special product pattern results in the following polynomial?
a. x2 + 6x + 9
b. x2 − 25
(x + 3)2
(x + 5)(x − 5)
c. x2 − 8x + 16
(x − 4)2
c. (x + 3)(x − 3)
y2
x2 − 9
f. (2x −1)(2x − 1)
4x2 − 4x + 1
Notes 9.3
3. Use a special products pattern to
find the product without a calculator: 19 • 21
19 • 21 = (20 − 1)(20 + 1)
= 400 − 1
= 399
4. Use a special products pattern to
find the product without a calculator: 212
212 = (20 + 1)2
= 400 + 40 + 1
= 441
5. In dogs, the gene E is for straight pointy ears and the gene e is for pointy but
droopy ears. Any gene combination with an E results in straight pointy ears
on a dog. The Punnett square shows the possible gene combinations of the
offspring and the resulting type of ear.
E
a. What percent of the possible gene combinations
of the offspring result in droopy ears?
E
25%
e
b. How can a polynomial model the possible
combinations of the offspring?
(0.5E + 0.5e)2 = 0.25E2 + 0.5Ee + 0.25e2
The coefficient of e2 shows that 25% of the possible
gene combinations result in droopy ears.
e
EE
Ee
Straight
Straight
Ee
ee
Straight
Droopy
Algebra Notes
Objective(s):
Section 9.4: Solve Polynomial Equations in Factored Form
To solve polynomial equations.
Vocabulary :
I. Zero-Product Property:
Let a and b be real numbers. If
II. Roots:
The solutions to ab = 0.
III. Factoring:
Writing a polynomial as a
product
polynomials
IV. Greatest Common
Monomial Factor (GCF): A
a • b = 0 then a = 0 or
of
b=0
other
.
monomial
with an
integer
evenly
into each of the polynomial’s terms.
coefficient that
divides
V. Projectile:
An object that is propelled into the air but has no power to keep itself in the air.
VI. Vertical Motion Model:
The height h (in feet) of a projectile can be modeled by the equation
(write this on your formula sheet)
.
h = −16t2 + vt + s
where t is the time (in seconds) the object has been
in the air, v is the initial vertical velocity (in feet per second), and s is the initial height
(in feet).
VII. To solve a polynomial
equation using the
zero-product property:
You may need to
factor
polynomials. Look for the
the polynomial, or write it as a product of other
GCF
Examples:
1. Solve each of the following.
a. (x + 3)(x − 5) = 0
−3 or 5
b. (2x + 1)(x + 4) = 0
− ½ or −4
2. Name the greatest common monomial factor of the polynomial.
a. 8xy + 20x
4x
b. 10x2y3 − 15xy
5xy
of the polynomial's terms.
Notes 9.4
3. Factor out the greatest common monomial factor.
a. 8x + 12y
b. 5x + 10y
c. 14x2 y2 + 21y4x3
4(2x + 3y)
5(x + 2y)
d. 8x3 + 10x4 + 2
e. 4x2y − 5xy + xy2
2(4x3 + 5x4 + 1)
7x2y2(2 + 3y2x)
f. 27x2y3 + 18x3y2 + 9
xy(4x − 5 + y)
4. Solve.
a. 3x2 + 18x = 0
3x(x + 6) = 0
x = 0 or x = −6
c. 4x2 = 14x
b. 4x2 + 2x = 0
2x(2x + 1) = 0
x = 0 or x = − ½
d. 6x2 = 15x
4x2 − 14x = 0
2x(2x − 7) = 0
6x2 − 15x = 0
3x(2x − 5) = 0
x = 0 or x = ⁷⁄₂
x = 0 or x = ⁵⁄₂
5. A dolphin jumped out of the water with an initial velocity of 32 feet per second.
After how many seconds did the dolphin enter the water?
h = − 16t2 + vt + s
h = −16t2 + 32t
0 = −16t2 + 32t
0 = −16t(t − 2)
t = 0 or t = 2
2 seconds
3(9x2y3 + 6x3y2 + 3)
Section 9.5: Factor x2 + bx + c
Algebra Notes
Objective(s):
To factor trinomials of the form x2 + bx + c.
Vocabulary :
Note: The method taught to you in this section only applies to a trinomial where
the leading coefficient is 1 (ex: x2 + 5x + 6).
You cannot use this method if the leading coefficient is not 1. ( ex: 4x2 + 8x + 3 )
I. Factoring x2 + bx + c:
x2 + bx + c =
(x + p)(x + q)
provided
p+q=b
and
p•q=c
Examples:
1. Which of the following trinomials can be factored using the method of this section. Circle all that apply.
A. x2 + 6x + 7
B. 6x2 + 7x + 1
D. 3x2 + 4x + 1
E. x2 − 3x − 4
C. 4x2 + 7x − 2
2. Factor each of the following.
a. x2 + 11x + 18
(x + 9)(x + 2)
d. x2 − 6x + 8
(x − 2)(x − 4)
g. x2 + 3x − 10
(x + 5)(x − 2)
b. x2 + 5x + 6
(x + 3)(x + 2)
e. x2 + 2x − 15
(x + 5)(x − 3)
c. x2 − 9x + 20
(x − 5)(x − 4)
f. x2 − 5x + 6
(x − 2)(x ‒ 3)
.
Notes 9.5
3. Factor each of the following.
a. x2 + 5x + 6
b. x2 − x − 6
(x + 3)(x + 2)
(x − 3)(x + 2)
c. x2 + x − 6
d. x2 − 5x + 6
(x + 3)(x ‒ 2)
(x − 3)(x − 2)
4. Study the factoring patterns in # 3.
a. What happens with the factors (p and q) when you have the “ + +” pattern (part a)?
Both p and q are positive numbers.
b. What happens with the factors (p and q) when you have the “− −” pattern (part b)?
One is positive the other is negative. The bigger number must be
negative.
c. What happens with the factors (p and q) when you have the “+ −” pattern (part c)?
One is positive the other is negative. The bigger number must be
positive.
d. What happens with the factors (p and q) when you have the “− +” pattern (part d)?
Both p and q are negative numbers.
5. Solve the equation.
a. x2 + 3x = 18
b. x2 − 2x = 24
(x +6)(x − 3) = 0
(x − 6)(x + 4) = 0
x = −6 or x = 3
x = 6 or x = −4
c. x2 = 3x + 28
(x − 7)(x + 4) = 0
x = 7 or x = −4
6. You are designing a flag for the school football team with
the dimensions shown in the diagram. The shaded region
will show the team name. The flag requires 117 square
inches of fabric. Find the width w of the flag.
w(w + 4) = 117
w2 + 4w − 117 = 0
w
 2" 
w+2
(w + 13)(w − 9) = 0
w = −13 or w = 9. w = −13 does not make sense with the problem.
Therefore, the width must be 9”.
Section 9.6: Factor ax2 + bx + c
Algebra Notes
Objective(s):
To factor trinomials of the form x2 + bx + c.
Vocabulary :
I. Two methods for factoring ax2 + bx + c:
1. Guess and check with factors of a and c:
Factor 2x2 − 7x + 3
Factors of 2
Factors of 3
1, 2
−1, −3
(x − 1)(2x − 3)
−3x − 2x = −5x
1, 2
−3, −1
(x − 3)(2x − 1)
−x − 6x = − 7x
Answer:
Possible Factorization
Middle term multiplied
(x − 3)(2x − 1)
2. Grouping method:
Factor 3x2 + 14x − 5
Step 1: Find two numbers
whose product is:
and whose sum is:
ac
b
product must be (−5)(3) = −15
and the sum must be 14
15 and −1 work. 15(−1) = −15
and 15 + −1 = 14
Step 2: Rewrite the middle term, 14x,
using the two numbers you found
in step 1. You will have a polynomial
with four terms.
Step 3: Group the first two terms and factor;
group the last two terms and factor.
There should be an common binomial
factor in each of these. Factor the
common binomial from each term.
3x2 + 14x − 5
3x2 + 15x − x − 5
3x(x + 5) − 1(x + 5)
(x + 5)(3x − 1)
II. Factoring when a is negative: To factor a trinomial of the form ax2 + bx + c when a is negative, first
factor ‒1
from each term of the trinomial. Then factor
the resulting trinomial using either guess and check or grouping.
Notes 9.6
Examples:
1. Factor any two of the following by guess and check and factor the other two by grouping.
a. 2x2 − 13x + 6
(x − 6)(2x − 1)
b. 4x2 − 12x − 7
c. 3x2 + 8x + 4
d. 4x2 − 9x + 5
(2x + 1)(2x − 7)
(x + 2)(3x + 2)
(x − 1)(4x − 5)
2. Factor each of the following using any method.
a. −4x2 + 12x + 7
−(2x +1)(2x − 7)
b. −3x2 − x + 2
c. −3x2 − 13x + 4
−(x + 1)(3x − 2)
−(3x + 1)(x + 4)
3. A soccer goalie throws a ball into the air at an initial height of 8 feet and an initial vertical velocity
of 28 feet per second.
a. Write an equation that gives the height (in feet) of the
soccer ball as a function of the time (in seconds) since
it left the goalie’s hand.
b. After how many seconds does it hit the ground?
h = − 16t2 + vt + s
h = − 16t2 + 28t + 8
0 = −4(4t2 − 7t − 2)
−4(4t + 1)(t − 2) = 0
t = − ¼ or t = 2
The ball hits the ground after 2 seconds.
4. A rectangle’s length is 5 feet more than 4 times the width.
The area is 6 square feet. What is the width?
w(4w + 5) = 6
4w2 + 5w − 6 = 0
(4w − 3)(w + 2) = 0
w = ¾ or w = −2

w = ¾ ft
Algebra Notes
Objective(s):
Section 9.7: Factor Special Products
To factor special products.
Vocabulary :
I. Difference of Squares Factoring Pattern:
(write this on your formula sheet)
a2 − b 2 =
(a + b)(a − b)
II. Perfect Square Trinomial Factoring Pattern:
(write this on your formula sheet)
a2 + 2ab + b2 =
a2 − 2ab + b2 =
(a + b)2
(a − b)2
Examples:
1. Factor each polynomial.
a. y2 − 9
b. 64x2 − 16
(y + 3)(y − 3)
d. 12 − 48x2
16(2x − 1)(2x + 1)
e. x2 + 6x + 9
12(1 − 2x)(1 + 2x)
(x + 3)2
g. 9m2 − 6my + y2
h. −2x2 − 16x − 32
(3m − y)2
−2(x + 4)2
2. Solve. x2 − 5x +
25
=0
4
(x − ⁵⁄₂)2 = 0
x = ⁵⁄₂
c. x2 − 81y2
(x − 9y)(x + 9y)
f. 4n2 + 20n + 25
(2n + 5)2
Notes 9.7
3. A rock is dropped from a riverbank that is 4 feet above the surface of the river.
After how many seconds does the rock hit the surface of the water?
−16t2 + 4 = 0
−4(4t2 − 1) = 0
−4(2t + 1)(2t − 1) = 0
t = − ½ second
4. A window washer drops a wet sponge from a height of 64 feet. After
how many seconds does the sponge land on the ground?
−16t2 + 64 = 0
−16(t2 − 4) = 0
−16(t − 2)(t + 2) = 0
t = 2 seconds
Algebra Notes
Objective(s):
Section 9.8: Factor Polynomials Completely
To factor polynomials completely.
Vocabulary :
I. Guidelines for Factoring a Polynomial Completely.
1. Factor out the
2. Look for a
(lesson 9.7)
greatest common monomial factor.
difference of two squares
3. Factor a trinomial of the form
factors
or a
(lesson 9.4)
perfect square trinomial
ax2 + bx + c
into a product of
grouping
.
binomial
.
4. Factor a polynomial with four terms by
Examples:
1. Factor the expression, if possible.
a. 4x(x − 3) + 5(x − 3)
(x − 3)(4x + 5)
b. 2y2(y − 5) − 3(5 − y)
2y2 (y − 5) + 3(y − 5)
(y − 5)(2y2 + 3)
c. x3 + 2x2 + 8x + 16
(x + 2)(x2 + 8)
e. x3 − 10 − 5x + 2x2
(x + 2)(x2 − 5)
g. 3x3 − 21x2 − 54x
3x(x + 2)(x − 9)
d. x2 + 4x + xy + 4y
(x + 4)(x + y)
f. x2 − 4x − 3
cannot be factored
h. 8x3 + 24x
8x(x2 + 3)
Notes 9.8
2. Solve.
a. 2x3 − 18x2 = − 36x
b. 3x3 + 18x2 = −24x
2x(x − 3)(x − 6) = 0
3x(x + 4)(x + 2) = 0
x = 0, 3, or 6
x = 0, −4, or −2
c. x3 − 8x2 + 16x = 0
d. x3 − 25x = 0
x(x − 4)(x − 4) = 0
x(x + 5)(x − 5) = 0
x = 0 or 4
x = 0, 5, or −5
3. A kitchen drawer has a volume of 768 in3.
The dimensions of the drawer are shown.
Find the length, width, and height of the drawer.
w(w + 4)(16 − w) = 768
−w3 + 12w2 + 64w − 768 = 0
−w2(w − 12) + 64(w − 12) = 0
(w − 12)(−w2 + 64) = 0
16 − w
w
w+4
w − 12 = 0 or −w2 + 64 = 0
w = 12
or w =  8
w = 12 or w = 8
Dimensions could be
or they could be
16 x 12 x 4
12 x 8 x 8