1

Algebra 2H Name:__________________________________ Lesson/HW- Factoring Polynomials Date:___________________________________ Objective: x 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

2

(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

3

Algebra 2H Name:__________________________________ Lesson/HW- Factoring Special Polynomials Date:___________________________________ Objective: x 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:

(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)

(3) Factor Trinomials – check for special products, for any numbers a and b: (a) Perfect Square Trinomials: a2 + 2ab + b2 = (a + b)2

a2 – 2ab + b2 = (a – b)2

(b) General Trinomials: acx2 + (ad + bc)x + bd = (ax + b)(cx + d)

(4) 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

4

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

5

Algebra 2H Name:__________________________________ Lesson/HW- Operations with Rational Expressions I Date:___________________________________ Objective: x 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) x12x4x20x4

20xx9x

2

2

2

2

��

x��� (6)

4x6x5x

6xx2x3x

2

2

2

2

���

y��

�

6

Showing all work, perform the indicated operation and simplify your answer.

(7) 12b

b4a12 3

x� (8) 22 x32

16x88x2

x8 �x

�

(9) x3x

18x15x36xx2

12x32

2

2

2

���

y��

� (10) 1x3x3

5x4x45x9

22 ��

y��

�

(11) 1x

1x2xx3xx

2

223

���

x� (12)

9x68x4

9x46xx2

2

2

��

y���

7

Showing all work, perform the indicated operation and simplify your answer.

(13) 72a9

a8a64a8

64a2

2

�x

�� (14)

1x2x3x3

x31x

22

2

���

x�

(15) 9c6

128c4

6cc2 2

�x

��� (16)

25x525x

5x25x10x 22

��

y�

��

(17) 1x

x28x2x

x8x10x222

23

�y

���� (18) � �9x4

x26xx2 2

2

2

�y��

8

Algebra 2H Name:__________________________________ Lesson/HW- Operations with Rational Expressions II Date:___________________________________ Objectives: x perform operations with algebraic fractions and simplify mixed expressions

x 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) 22 y168

16yy2

��

� (6)

k43k8

3k42k7

��

���

9

Showing all work, perform the indicated operation and simplify your answer.

(7)

x11

x1x

�

� (8)

65

3x

x52

�

�

(9) x1

xx1x2

2 ��� (10)

n239

3n2n4 2

��

�

10

Showing all work, perform the indicated operation and simplify your answer.

(11) 1a

11a1a3

2 ��

�� (12)

42y

54y3 �

��

(13) 4

44 ��

� xx

x (14) 4x

216x

x2

2

��

�

11

Showing all work, perform the indicated operation and simplify your answer.

(15) 3a

16a5a

5a22 �

���

� (16) 4

112

122 �

����

bbbb

(17)

x13

31x

�

� (18)

x1

z1

xz

zx

�

�

12

Algebra 2H Name:__________________________________ Lesson/HW- Complex Fractions and Equations Date:___________________________________ Objectives: x simplify mixed expressions and complex fractions

x solve equations with algebraic expressions x 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:

(1) ab

bba

a 22

��

� (2)

y1y1y �

�� (3)

1m3m5mm

���

�

(4)

y31

y6

y51 2

�

��

Solve each of the following equations and check:

(5) 8x

164x

168x

x2 �

�

��

(6) 1h

61h

11h

12 �

�

��

(7) 9b

46b2

16b2

12 �

�

��

(8) 16x

164x

18x2

x2 �

�

��

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.

(10) If the length of a rectangular field is represented by the expression 9aaa3

2

2

�� , and the width is represented

by 4a

12aa2

��� , what simplified expression represents the area of the field?

13

Unit 1: Algebraic Fractions, Equations & Factoring

Definitions, Properties & Procedures

Factoring the process of writing a number or algebraic expression as a product

Least Common Denominator (LCD) the least common multiple of two or more given denominators

Algebraic Fraction has the same properties as a numerical fraction, only the numerator and denominator are both algebraic expressions

Rational Expression

an algebraic expression whose numerator and denominator are polynomials and whose denominator has a degree of one or greater

Simplifying Rational

Expressions

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!

Multiplying & Dividing Rational

Expressions

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

Adding & Subtracting

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

LCD; and multiply each numerator by the same factor that you multiplied its denominator by (multiply by a “fraction of one”)

(3) Combine all numerators (make sure the signs are placed appropriately) and simplify; and put over LCD

(4) Reduce if possible

Complex Fraction

a fraction that contains one or more fractions in the numerator, the denominator, or both To simplify complex fractions: 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.

Rational Equation

an equation that contains one or more rational expressions To solve rational equations: (1) Find the LCD (2) Multiply each fraction by this LCD (3) Cancel all denominators (4) Solve the remaining equation for the given variable

Greatest Common Factor (GCF)

the product of the greatest integer and the greatest power of each variable that divides evenly into each term

14

Difference of Two Perfect Squares

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 Æ ( )( ) (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

& Difference of Two

Perfect Cubes

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 Æ ( )( ) (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: ( + )( í������������� 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: )ROORZ�DERYH�VWHSV�DQG�DUUDQJH�WKH�VLJQV�DV�IROORZV�������í������������+ )

Perfect Square Trinomial

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 Æ ( )( ) (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

Factoring by Grouping

(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

Factoring Trinomials with a

Leading Coefficient

Greater Than One

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

15

Algebra 2H Name:__________________________________ Review- Rational Expressions Test 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

(2) a2b2 + ab – 6

(3) t2 – 2t + 35

(4) y4 – z2

(5) x5 + 27x2

(6) x4 – 81

(7) 3d2 – 3d – 5

(8) 72 – 26y + 2y2

(9) x3 + 7x2 + 2x + 14

Perform the indicated operation and simplify your answer.

(10) 2

2

2

2

a61a9

1a6a9a2a6 �

x��

�

(11) 12t7t20tt

25t10t9t6t

2

2

2

2

����

x����

(12) 45x5

12x33x7x2

4x22 ��

y��

�

(13) 4x

6x5x6xx2

x3x2

2

2

2

���

y��

�

(14) a2

43a

7�

��

(15) x1

1x1

��

(16) m9

189m

m2�

��

(17) 22 yx

6yx

3yx

3

�

��

�

(18)

3x3

3x1

2

�

�

(19)

2y31

2y71

��

��

(20)

1x241x

1x41

���

��

Answer the following word problems, showing all work to explain your answer: (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.

(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.

Solve each of the following equations and check:

(23) 4x3

2x25

1x4

��

�

(24) 8a2a

a212a

14a

22 ��

�

��

�

16

Algebra 2H Mixed Practice with Algebraic Fractions & Equations Name: Objective: x 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!

Why did 51

go to a psychiatrist?

22

22

yxyx2xy

xyx2y

���

x��

x8x28x

xx34040x8

22 ��

x��

�

� �25x55x16x3

xx92

3

�y��

�

4x1

12xx7

3x2

2 ��

���

�

� � ¸̧¹

·¨̈©

§

��

y���

y� 232

22

x2x55x

x7xxx23512x75

x96

27x3x2

��

�

1x1

x3

xx5x

2 � �

��

3x122

3xx4

��

�

8x2xx21

2x1

4x2

2 ���

�

��

3x1

9x6

3xx

2 �

��

�

5x(3x + 1)

-1

1 and 3

3 3x + 12

1

No Solution

EH

BE

US

EN

CA

AS

TO

SE

EW

OT

17