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Algebra 2H
Lesson/HW- Factoring Polynomials
Name:__________________________________
Date:___________________________________
Objective:
factor polynomials
Factor completely. If the polynomial cannot be factored, write simplified.
(1)
8x – 24
(2)
xy – 17y
(3)
x2 – 169
(4)
x2 – y2
(5)
x2 + y2
(6)
3x3 – 3x
(7)
9x2 – 36y2
(8)
2x3 – 4x2 – 6x
(9)
5x2 – 13x + 6
1
(10) x2 – 6x + 2
(11) 4a2 + 12ab + 9b2
(12) 36w2 – 16
Factor completely. If the polynomial cannot be factored, write simplified.
(13) 6 – 5x + x2
(14) 40 – 76x + 24x2
(15) 2x2 + 4x – 1
(16) 2x2 + 28x – 30
(17) 6x2 + 7x – 3
(18) 18x2 – 31xy + 6y2
2
Algebra 2H
Lesson/HW- Factoring Special Polynomials
Name:__________________________________
Date:___________________________________
Objective:
factor special polynomials
Guidelines for Factoring:
(1)
Factor out the Greatest Common Factor (GCF) – “Gotta Come First” Monomials
(2)
Factor Binomials – check for special products, for any numbers a and b:
(3)
(a)
Difference of Two Perfect Squares:
a2 – b2 = (a + b)(a – b)
(b)
Sum of Two Perfect Cubes:
a3 + b3 = (a + b)(a2 – ab + b2)
(c)
Difference of Two Perfect Cubes:
a3 – b3 = (a – b)(a2 + ab + b2)
Factor Trinomials – check for special products, for any numbers a and b:
(a)
a2 + 2ab + b2 = (a + b)2
Perfect Square Trinomials:
a2 – 2ab + b2 = (a – b)2
(b)
(4)
acx2 + (ad + bc)x + bd = (ax + b)(cx + d)
General Trinomials:
Factor Polynomials – if there are four or more terms, try factoring by grouping.
Factor completely:
(1)
x3 + 27
(2)
x3 – 64
(3)
27x3 – 8
(4)
2x3 + 16
(5)
x3 – 4x2 + 3x – 12
(6)
x3 + 5x2 – 2x – 10
(7)
5a2x + 4aby + 3acz – 5abx – 4b2y – 3bcz
3
Factor completely:
(8)
35x3y4 – 60x4y
(9)
2r3 + 250
(10) 100m8 – 9
(11) 3z2 + 16z – 35
(12) 162x6 – 98
(13) 4m6 – 12m3 + 9
(14)
x3 – 343
(15) ac2 – a5c
(16) c4 + c3 – c2 – c
(17) ax – ay – bx + by
(18) 64x3 + 1
(19) 3ax – 15a + x – 5
4
Algebra 2H
Lesson/HW- Operations with Rational Expressions I
Name:__________________________________
Date:___________________________________
Objective:
perform operations with algebraic fractions and simplify mixed expressions
Do Now: Factor completely. If the polynomial cannot be factored, write simplified.
(1)
3x2 + 10x + 8
(2)
8x3y6 + 27
(3)
4x2 – 12x + 5
(4)
4x2 – 4x – 48
Showing all work, perform the indicated operation and simplify your answer.
(5)
x2
x
2
9
x 20
4x 2
4x 2
20 x
12x
(6)
x 2 3x
2x 2 x 6
x2
5x 6
x
4
2
5
Showing all work, perform the indicated operation and simplify your answer.
b3
12
(7)
12a 4
b
(9)
3 x 2 12
2x 2 x 6
(11)
x3
x2
3x
x2
(8)
3x 2
15 x 18
x
3x
2
2x 1
2
x 1
(10)
(12)
8x
2x 2 8
9x
x
2
45
4x 5
2x 2 x 6
4x 2 9
8 x 16
32x 2
3x
x2
3
1
4x 8
6x 9
6
Showing all work, perform the indicated operation and simplify your answer.
(13)
a2
8a2
(15)
2c 2 c 6
4c 8
(17)
2x 3 10 x 2 8 x
x 2 2x 8
64
64a
8a
9a 72
12
6c 9
2x
x
2
1
(14)
x2 1
3x 2
(16)
x 2 10 x 25
x 5
(18)
2x 2
3x 3
x 2x 1
2
x 6
2x 2
x 2 25
5 x 25
4x 2
9
7
Algebra 2H
Lesson/HW- Operations with Rational Expressions II
Name:__________________________________
Date:___________________________________
Objectives:
perform operations with algebraic fractions and simplify mixed expressions
simplify mixed expressions and complex fractions
Do Now: Factor completely. If the polynomial cannot be factored, write simplified.
(1)
x2 + 2x + xy + 2y
(2)
64x2 – 676
(3)
1 – 125y3
(4)
3a2 – 2b – 6a + ab
Showing all work, perform the indicated operation and simplify your answer.
(5)
2y
y 16
2
8
16
y
2
(6)
7k
4k
2
3
8 k
3 4k
8
Showing all work, perform the indicated operation and simplify your answer.
(7)
1
x
1
1
x
(9)
2x 1
x2 x
2
x
(8)
1
x
(10)
x
3
5
x
5
6
4n 2
2n 3
9
3 2n
9
Showing all work, perform the indicated operation and simplify your answer.
(11)
(13)
3a 1
a2 1
x
4
x
1
a 1
x
4
4
(12)
(14)
3y 4
5
y
2
4
x2
x
2
2
16
x
4
10
Showing all work, perform the indicated operation and simplify your answer.
(15)
2a 5
a
5a 6
2
x
(17)
3
1
3
1
x
1
a 3
(16)
(18)
2b 1
b b 12
2
x
z
1
z
1
b
4
z
x
1
x
11
Algebra 2H
Lesson/HW- Complex Fractions and Equations
Name:__________________________________
Date:___________________________________
Objectives:
simplify mixed expressions and complex fractions
solve equations with algebraic expressions
solve real-world applications with algebraic expressions
ON A SEPARATE SHEET OF PAPER,
ANSWER EACH OF THE FOLLOWING QUESTIONS SHOWING ALL WORK!
Perform the indicated operation and simplify your answer:
2
(1)
a
a b
2
b
b a
(2)
y 1
y 1
y
1
m 5
m
m 3
m 1
(3)
(4)
5
y
1
6
y2
3
y
Solve each of the following equations and check:
(5)
(6)
x
x
8
16
x
64
1
h 1
1
2
x
1
h 1
6
h
2
1
8
(7)
1
2b 6
1
2b 6
(8)
x
2x 8
1
x
4
4
b
2
9
16
x 16
2
Show All Work:
(9)
The area of a rectangular patio is represented by the expression (6x2 + 13x – 5). The width of the patio is
(3x – 1). Write a simplified expression to represent the length of the patio in terms of x.
3a a 2
(10) If the length of a rectangular field is represented by the expression 2
, and the width is represented
a 9
a 2 a 12
by
, what simplified expression represents the area of the field?
a 4
12
Unit 1: Algebraic Fractions, Equations & Factoring
Definitions, Properties & Procedures
Factoring
the process of writing a number or algebraic expression as a product
Least Common
the least common multiple of two or more given denominators
Denominator (LCD)
Algebraic Fraction
Rational
Expression
Simplifying
Rational
Expressions
Multiplying &
Dividing Rational
Expressions
has the same properties as a numerical fraction, only the numerator and
denominator are both algebraic expressions
an algebraic expression whose numerator and denominator are polynomials and
whose denominator has a degree of one or greater
reducing or simplifying a rational expression means to write the expression in
lowest terms, which can only be done with a single fraction, a product of fractions
or a quotient of fractions. If there is an addition or subtraction sign in the
numerator (or denominator), it must be factored first and then like factors with the
denominator (or numerator) can be canceled. Note: you cannot reduce across a
sum or difference of two or more fractions!
To multiply rational expressions:
(1) Factor each numerator and denominator completely
(2) Cancel any like factors in any numerator with any like factors in any
denominator
(3) Multiply the remaining expressions in each numerator
(4) Multiply the remaining expressions in each denominator
(5) Reduce if possible
To divide rational expressions:
(1) Multiply the first fraction by the reciprocal of the second fraction (KCF)
(2) Follow the steps above to multiply rational expressions
(1) Find the least common denominator among all fractions (if necessary)
(2) Multiply each denominator by an appropriate factor to make it equivalent to the
Adding &
LCD; and multiply each numerator by the same factor that you multiplied its
Subtracting
denominator by (multiply by a “fraction of one”)
Rational
(3) Combine all numerators (make sure the signs are placed appropriately) and
Expressions
simplify; and put over LCD
(4) Reduce if possible
a fraction that contains one or more fractions in the numerator, the denominator, or
both
To simplify complex fractions:
Complex Fraction
Combine fractions in the numerator and denominator separately by adding or
subtracting. Once there is a simplified fraction above a fraction, use the steps for
dividing fractions to further simplify the expression.
an equation that contains one or more rational expressions
To solve rational equations:
(1) Find the LCD
Rational Equation
(2) Multiply each fraction by this LCD
(3) Cancel all denominators
(4) Solve the remaining equation for the given variable
Greatest Common the product of the greatest integer and the greatest power of each variable that
Factor (GCF) divides evenly into each term
13
Difference of Two
Perfect Squares
Sum of
Two Perfect Cubes
&
Difference of Two
Perfect Cubes
Perfect Square
Trinomial
Factoring by
Grouping
Factoring
Trinomials with a
Leading
Coefficient
Greater Than One
a polynomial of the form a2 – b2, which may be written as the product (a +
b)(a – b)
To factor a difference of two perfect squares:
(1) Create two empty binomials
(
)(
)
(2) Take the square root of the first term of the given binomial and put it in the 1st
position in each binomial
(3) Take the square root of the last term of the given binomial and put it in the 2nd
position in each binomial
(4) Make one binomial a sum and the other binomial a difference
Sum of Two Perfect Cubes: a polynomial of the form a3 + b3, which may be
written as the product (a + b)(a2 – ab + b2)
To factor a sum of two perfect cubes:
(1) Create an empty binomial and an empty trinomial (
)(
)
(2) Take the cube root of the first term of the given expression
(a) put it in the 1st position in the binomial
(b) square it and put it in the 1st position of the trinomial
(3) Take the cube root of the last term of the given expression
(a) put it in the 2nd position in the binomial
(b) square it and put it in the last position of the trinomial
(4) Find the product of the terms in the binomial and put it in the middle position
of the trinomial
(5) Arrange the signs as follows: ( + )(
Difference of Two Perfect Cubes: a polynomial of the form a3 – b3, which
may be written as the product (a – b)(a2 + ab + b2)
To factor a difference of two perfect cubes:
+ )
a trinomial whose factored form is the square of a binomial; has the form
a2 – 2ab + b2 = (a – b)2 or a2 + 2ab + b2 = (a + b)2
To factor a perfect square trinomial:
(1) Create two empty binomials
(
)(
)
(2) Take the square root of the first term of the given trinomial and put it in the 1st
position in each binomial
(3) Take the square root of the last term of the given trinomial and put it in the 2nd
position in each binomial
(4) The signs of each binomial should be the same as the middle term of the given
trinomial
(1) Find a convenient point in the polynomial to partition (or group)
(2) Factor within each group
(3) Factor out the Greatest Common Factor across the groups
To factor trinomials in the form ax2 + bx + c:
(1) Multiply the a term by the c term
(2) Find the factors of (ac) which will add to the b term
(3) Rewrite the b term as the sum of two x terms with coefficients being the
factors of (ac)
(4) Group the first two terms and last two terms each in a set of parentheses
(5) Factor out the Greatest Common Factor from each group
14
Algebra 2H
Review- Rational Expressions Test
Name:__________________________________
Date:___________________________________
ANSWER EACH ON A SEPARATE SHEET OF PAPER. SHOW ALL WORK!
Factor completely. If the polynomial cannot be factored, write simplified.
(1)
6c2 + 13c + 6
(4)
y4 – z2
(7)
3d2 – 3d – 5
(2)
a2b2 + ab – 6
(5)
x5 + 27x2
(8)
72 – 26y + 2y2
(3)
t2 – 2t + 35
(6)
x4 – 81
(9)
x3 + 7x2 + 2x + 14
Perform the indicated operation and simplify your answer.
(10)
6a 2 2a
9a 2 6a 1
9a 2 1
6a 2
(14)
7
4
a 3
2 a
1
(18)
3
(11)
t2
t2
6t 9
10t 25
t 2 t 20
t 2 7t 12
(15)
1
x
1
1 x
7
1
(12)
(13)
x
2x 2
4
7x
3
x 2 3x
2x 2 x 6
3 x 12
5 x 2 45
x2
5x 6
2
x
4
(16)
2m
m 9
x
y
(19)
18
9 m
3
(17)
x
3
x2
3
1
y
3
y
x
y
1
x2
2
4
x 1
24
x 1
x 1
(20)
6
2
3
y2
Answer the following word problems, showing all
work to explain your answer:
Solve each of the following equations and check:
(21) The area of a rectangle is (x2 – x – 6) square
meters. The length and width are each
increased by 9 meters. Write the area of the
new rectangle as a trinomial in terms of x.
(23)
(22) The freshman and sophomore classes both
participated in a fundraiser. The freshman
class collected (4x2 – 1) and the sophomore
class collected (6x2 + 7x + 2). Express, in
simplest form, the ratio of the sophomore’s
collection to the freshman’s collection.
(24)
4
x 1
5
2x 2
2
a
1
4
a
2
3x
4
1 2a
a
2a 8
2
15
Algebra 2H
Mixed Practice with Algebraic Fractions & Equations
Objective:
Name:
simplify rational expressions and solve equations with fractional algebraic expressions
For each of the following, perform the indicated operation and simplify
your answer, or solve the equation. Complete on a SEPARATE
SHEET OF PAPER SHOWING ALL WORK!
y2
EH
y 2x
y x
BE
8 x 40
40 3 x x 2
US
9x 3 x
3 x 2 16 x 5
EN
CA
2x 2
2
x
3
75 x 2
x
12
2
Why did
1
5
go to a psychiatrist?
x2
xy y 2
AS
2x
3 x 27
x 8
2x 2 8 x
TO
x 5
x2 x
3
x
SE
4x
x 3
2
5x
25
6
9
2
7
x 12
EW
1
x
35 2x x
x 2 7x
x
4
1
x 1
12
x 3
1
4
x
x
5x
3
OT
5
2x 2
x
1 2x
x
2x 8
2
2
x
2
x
6
3
x
2
1
9
x
3
No Solution
1
3x + 12
3
1 and 3
-1
5x(3x + 1)
16
17