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CHAPTER
5A: POLYNOMIALS
Algebra 2A -2012
LESSON 5.1A

I can identify the base and the exponent.

I can simplify product of monomials.
What is it?
Examples:
n
b
Power
Important characteristic:
Base: ____
Exponent: ____
Shortcut for repeated multiplications
Nonexamples:
Example 1: Simplify the following
5)(42)
(4
a.
b.
(6x2)(x4)
4y)(-4x2)
(3x
c.
Your Turn 1: Simplify the following
1.
(23)(24)
2.
(5v4)(3v)
6)(7a2b3)
(-4ab
3.
What is it?
Examples:
bn • bm = bn + m
same bases
Important characteristic:
Add the
exponents
x3 • x4
(2x3y)(5x)
Product
of Powers Nonexamples:
x3 • x4 ≠ x12
o I can simplify product of Powers.
Example 2: Simplify the following
a.
(x5)2
b.
(y7)3
c.
(3xy)2
Your Turn 2: Simplify the following
1.
(m 6)2
2.
(2b2)4
2
(4abc)
3.
Example 3: Simplify the following
d.
(10ab4)3 (3b2)2
Your Turn 3: Simplify the following
4.
(2xy2)3 (-4x5)2
o I can simplify quotient of monomials.
Example 4
a.
16x3y4
4 x5y
3a b 
7 2 2
b.
12a 4b8
Your Turn!
(3xy5)2
(2x3y7)3
Homework: Lesson 5.1 A
Check your answers !
LESSON 5.1B

I can simplify quotient of monomials.
I can simplify monomials with negative exponents.
 I can simplify monomials with a zero exponent.

(5xy)0 = 1
(b)0 = 1
Zero
Exponents
Anything to the
power of zero is one!
8-2
(5xy)0 ≠ 0
(b)-n
(5a)-1
1
= bn
1
=
5a
Negative
Exponents
Negative exponents
“move” up or down to
make it a positive exp.
8-2
(5a)-1 ≠ -5a
NEGATIVE EXPONENT RULE:
a-n
1
n
a
=
-n
n
a is the reciprocal of a
Example1:
1.
x-3
2.
5a-2b0
3.
-4a-7b-3
Your Turn 1:
a.
3w-3
b.
4x-7
c.
3x0y-4
Example 2:
4.
0
6a
2
3a
5.
5 7
4a b
 2 3
6a b
Your Turn 2:
d.
2
5
9
y
e.
0
2
3x y
4
15 x y
f.
 12ab 
  4  4 
 4a b 
3
0
Learning target(s):
 I can multiply and divide monomials. (LT1)
Mixed practice:
Simplify expressions means to write expressions without parentheses or
negative exponents.
a.
b.
c.
Learning target(s):
 I can multiply and divide monomials. (LT1)
Mixed practice:
d.
e.
f.
Learning targets:
 I can multiply and divide monomials. (LT1)
Mixed practice:
g.
h.
Learning target(s):
 I can multiply and divide monomials. (LT1)
Mixed practice:
Simplify expressions means to write expressions without parentheses or
negative exponents.
a.
b.
c.
Learning target(s):
 I can multiply and divide monomials. (LT1)
Mixed practice:
d.
e.
f.
Learning targets:
 I can multiply and divide monomials. (LT1)
Mixed practice:
g.
h.
Homework: Lesson 5.1B
Warm- up over Lesson 5.1A & B
Simplify the following.
 4x 


3 
 3 xy 
2
1.
2
x x x
8
5
2.
2
4

3
3.
4.
5.
6. Write a problem involving operations with powers whose answer is: -8x3
5.1B
LESSON 5.2 (LT3)
I can recognize a monomial, binomial, or trinomial.
 I can identify the degree of a polynomial.
 I can rewrite polynomials in descending order.


I can add polynomials.

I can subtract polynomials.

I can multiply polynomials.
DEFINITION OF MONOMIAL
Example 1a:
CLASSIFY AS MONOMIAL OR NON-MONOMIAL
Examples:
Non Examples:
DEFINITION OF POLYNOMIALS
Example 1b:
CLASSIFY AS POLYNOMIAL OR NON-POLYNOMIAL
Examples:
Non Examples:
51
Your Turn 1:
Is it a polynomial? Yes or No?
Classify as monomial,
binomial, or trinomial?
a.
2x3 + 4x2
4
b.
2xy3 – 4x4y0 + 2x
c.
x2 + 3xy
y
d.
4x-2
Example 2: Arrange the terms of the polynomial so that the
powers of x are in descending order
Descending:
decreasing (biggest to smallest)
a.
4x2 + 7x3 + 5x
b.
9x3y – 4x5 + 8y - 6xy4
Your turn 2:
c.
10 + 7x3y – 2x4y2 + 8x
Example 3: Find the degree of the monomial.
Degree of a monomial: sum of the exponents of the variables
a.
5x2
b.
-9x3y5
Your Turn 3:
c.
7xy5z4
d.
10
Example 4: Find the degree of the polynomial.
Degree of a polynomial: it is the highest degree after
finding the degree of each term.
a.
5x4 + 3x2 – 9x
Your Turn 4:
b.
4x3 – 7x + 5x6
Check for Understanding Lesson 5.2 A: (LT3)
Is the polynomial a monomial, binomial, or trinomial.
a. y3 – 4
b. 3y3 + y0 – 2x
c. -4xy5
Arrange the terms of the polynomial so that the powers of x are in
descending order.
d. 3x4y3 – x6y0 + 6 – 2x
Find the degree of the monomial
e.
-2x4y3z
f.
Find the degree of the polynomial.
g. 3x4y3 – x6y0 + 6 – 2x
3ab0y3
Example 5: Simplify the following
1.
(x2 – 5x + 3) + (2x2 + 4x – 5)
2.
(5y2 – 3y + 8) + (4y2 – 9)
Your Turn 5: Simplify the following (LT4)
(3x2 – 4x + 8) + (2x -7x2 -5)
Example 6: Simplify the following
3.
(2x2 – 3x + 1) – (x2 + 2x – 4)
4.
(4y2 – 9) – (5y2 – 3y + 8)
Your Turn 6: Simplify the following (LT4)
(3x2 – 4x + 8) – (2x -7x2 -5)
Example 7: Simplify the following.
1.
y( 7y +
9y2)
2.
-3x(2x2 + xy – 3)
Your Turn 7: Simplify the following.
1.
n2(3n2 + 7)
2.
n2(3n2p - np + 7p)
Example 8:
1.
3.
Multiply the following.
(x + 3)(x + 2)
2.
(x  9)(3x + 7)
Your Turn 8: Find the product. (LT5)
a.
b.
Simplify.
c.
d.
e.
Homework:
Practice 5.2
Mixed Review Lessons 5.1 & 5.2:
Simplify the following
1.
(-2a2 b0)3 =
2.
3x2y(-4x + 2y3) =
3.
(2x2 – 3x – 10) – (3x + 4x2 – 4)
Find the degree of the polynomial.
4.
-52x2yz4 – 7x4y3z
Mixed Review Lessons 5.1 & 5.2
Simplify the following.
5.
(3x + 2y) + (3x – 2y)
7.
4x(3x + 2y)
(3x + 2y) – (3x – 2y)
6.
8.
(3x + 2y)(3x – 2y)
Mixed Review Lessons 5.1 & 5.2
Simplify the following
9.
(x3 y 2)(x 4 y6) =
10.
(32x5 y8) 2 =
11. (4x 2 – 8x + 10) – (x 2 – 12)
12.
13.
Find the degree of the polynomial.
14.
3x2y5 – 10xy8
Simplify the following.
15.
5
x x
 2.38 1013 

16. 
18 
 4.2 10 
0
17. 3  2 x   5x  7
18.
3  2x5x  7
2
(
2
x

5
)
19.
20. Write a problem involving operations with polynomials whose answer is:
9x2 – 30x + 25
DIVIDING POLYNOMIALS
LESSON
5-3 A
Learning targets:
I can divide polynomials by a monomial.(LT6)
I can divide polynomials using long division. (LT7)
Notes first!
We will start the Quiz at
Expressions that represent division:
Non-examples:
To divide a polynomial by a monomial,
use the properties of powers from lesson 5-1.
Example 1: Simplify.
Example 2:
Simplify
Your Turn 1: Simplify
Review: How to divide using long division.
1313
5
To divide by a polynomial by a polynomial,
use a long division pattern.
Remember that only like terms can be added or subtracted.
Example 3: Use long division to find the following.
Example 4: Use long division to divide.
EXAMPLE 5:
Your Turn 2: Divide using long division.
Your Turn 2: Divide using long division.
Your Turn 2: Divide using long division.
CLOSURE LESSON 5.3A: DIVIDING POLYNOMIALS
Homework: Practice 5.3 part I
CLOSURE LESSON 5.3A: DIVIDING POLYNOMIALS
Homework: Practice 5.3 part I
Warm- up Lesson 5.3a
Simplify the following.
1.
6n 3 y
2n 1 y 3
2
2
3b
(4b

7)

2b(b
 5b  3)
2.

3
2
3.  3x (5 x 5 x  3)
4. (3ab3c  9bc 2  12a 3b5c)  (3abc) 1
5. Write a problem involving division of polynomials whose answer is:
x-3
Part I
DIVIDING POLYNOMIALS
LESSON
5-3 B
Learning targets:
I can divide polynomials using synthetic division. (LT8)
Use Synthetic Division: A procedure to divide a
polynomial by a binomial using coefficients of the
dividend and the value of r in the divisor x – r.
Example 6: Use synthetic division to divide.
Use Synthetic Division: A procedure to divide a polynomial
by a binomial using coefficients of the dividend and the
value of r in the divisor x – r.
Example 7: Use synthetic division to divide.
Your Turn 3: Divide using synthetic division.
Closure Lesson 5-3: Dividing
Polynomials
Homework: Practice 5.3 part II (worksheet)
Closure Lesson 5-3: Dividing
Polynomials
Homework: Practice 5.3 part II (worksheet)
Warm- up Lesson 5.3 part II
1. (4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division
2. (x3 - 4x2 + 6x – 4) ÷ (x – 2) synthetic
Part II
Warm- up Lesson 5.3 part II
Alg. 2B
1. (4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division
2y – 1
4y4 + 0y3 – 5y2
+ 2y + 4
2. (x3 - 4x2 + 6x – 4) ÷ (x – 2) synthetic
Part II
FACTORING POLYNOMIALS
LESSON 5-4
A
Learning Targets:
I can factor polynomials using GCF.
I can factor polynomials with four terms by
grpouping.
I can factor trinomials.
I can factor polynomials with two terms.
Technique
It means…
GCF
Grouping
Trinomials
PST
Difference of
squares
Sum of Cubes
Difference of
Cubes
Example…
FACTORING BY GCF
I
can factor a polynomial by using Greatest
Common Factor.
Simplifying
Factoring
3(a + b) =
3a + 3b =
x(y – z) =
xy – xz =
6y(2x + 1) =
12xy + 6y =
Example 1: Factor by using Greatest Common Factor (GCF).
Find the GCF of each term.
Use the GCF to find the remaining factors.
Your Turn 1: Factor by using Greatest Common Factor (GCF).
Find the GCF of each term.
Use the GCF to find the remaining factors.
Example 2: Factor by using Greatest Common Factor (GCF).
4x2y – 10x3y4
Your Turn 2: Factor by using Greatest Common Factor (GCF).
14ab3 – 8b3
Example 3: Factor by using Greatest Common Factor (GCF).
2x4y2 – 16xy + 24x3y3
Your Turn 3: Factor by using Greatest Common Factor (GCF).
15a3b2 + 10a2b4 – 25ab3
FACTORING BY GROUPING
I
can factor polynomials with FOUR terms by
grouping .
Example 4: Factor by grouping.
Example 5: Factor by grouping.
Your Turn 4: Factor the following by grouping or using the area model.
Your Turn 5: Factor the following by grouping or using the area model.
WARM UP:
Simplify:
1) (x3 – 2x2 + 2x – 6) ÷ (x - 3) use synthetic division
2) (3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) use long division
Factor:
3) 7xy3 – 14x2y5 + 28x3y2 using GCF
4) ab – 5a + 3b -15 using grouping
WARM UP/REVIEW
Simplify:
1) (x3 – 2x2 + 2x – 6) ÷ (x - 3) use synthetic division
2) (3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) use long division
Factor:
3) 7xy3 – 14x2y5 + 28x3y2 using GCF
4) ab – 5a + 3b -15 using grouping
Review: Simplify
5)
(x – 3y)(x + 3y)
6)
(y – 2)(y + 5)
7)
(5 + 2x) + (-1 – x)
8)
(-7 – 3x) – (4 – 3x)
WARM UP DAY 6-ANSWERS
Simplify:
1) (x3 – 2x2 + 2x – 6) ÷ (x - 3) x2 + x + 5 + 9/(x-3)
2) (3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) x3 + x2 - 2x + 2 + 1/(3x-2)
Factor:
3) 7xy3 – 14x2y5 + 28x3y2 7xy2(y – 2xy3 + 4x2)
4) ab – 5a + 3b -15 (a+3)(b – 5)
Review: Simplify
5) (x – 3y)(x + 3y) x2 – 9y2
6) (y – 2)(y + 5)
y2 + 3y - 10
7) (5 + 2x) + (-1 – x) 4 + x
8) (-7 – 3x) – (4 – 3x) -11
Part I
FACTORING TRINOMIALS
LESSON
I
5-4 A
can factor a trinomial.
Technique
It means…
GCF
Grouping
Trinomials
PST
Difference of
squares
Sum of Cubes
Difference of
Cubes
Example…
Find factors (product) of ____________
whose __________ is __________ .
Example 1: Factor the trinomial.
a.
b.
Your Turn 1: Factor the trinomial.
a.
b.
Find factors (product) of
REPLACE
a & c whose sum is b
.
bx using those two numbers and factor by grouping.
Example 2: Factor the trinomials.
a.
Example 3: Factor the trinomials.
b.
Your Turn 2: Factor the trinomials.
a.
b.
If three terms: Factoring Trinomials
Example 3: Factor completely. If not possible, write prime.
a)
b)
Your Turn 2: Factor completely. If not possible, write prime.
a)
b)
c)
d)
e)
f)
g)
h)
WARM-UP/REVIEW LESSON 5.1- 5.4:
I. Factor completely (lesson 5.4):
1) 40xy + 30x – 100y – 75
2)
3y2 + 21y +36
II. Simplify (lessons 5.1, 5.2 & 5.3):
3) –5(2x)4
4) x5 • x-12 • x
5) (15q6 + 5q2)(5q2)-1
III. Simplify by long division or synthetic: (it’s your choice!)
6)
(x3 + 7x2 + 14x + 3) ÷ (x + 2)
Part II
WARM-UP/REVIEW LESSON 5.1- 5.4:
I. Factor completely (lesson 5.4):
1) 40xy + 30x – 100y – 75
5(4y + 3)(2x – 5)
2)
3y2 + 21y +36
3(y + 4)(y + 3)
II. Simplify (lessons 5.1, 5.2 & 5.3):
3) –5(2x)4
4) x5 • x-12 • x
-80x4
1
x6
5) (15q6 + 5q2)(5q2)-1
3q4 + 1
III. Simplify by long division or synthetic: (it’s your choice!)
6)
(x3 + 7x2 + 14x + 3) ÷ (x + 2)
x2 + 5x + 4 + -5
x+2
FACTORING POLYNOMIALS
LESSON
5-4
Objectives:
Factor polynomials with three or two terms
Identify perfect square trinomials (PST), difference of two squares,
& sum or difference of cubes
Go to pg. 20
Special Case of trinomial: PST
Example 4:
a)
b)
If two terms:
I) Difference of two squares
Example 5:
a)
c)
b)
d)
If two terms:
II) Sum of two cubes
Example 6:
If two terms:
III) Difference of two cubes
Example 7:
YOUR TURN 3: FACTOR COMPLETELY
4f2 – 64
a.
c.
n3 – 125
YOUR TURN 3: FACTOR COMPLETELY
4f2 – 64
a.
Difference of two squares;
4(f + 4)(f – 4)
c.
n3 – 125
Difference of two cubes;
(n – 5)(n2 + 5n + 25)
Simplify quotients of polynomials by factoring
Assume no denominator is zero.
Example
Your Turn
Lesson 5-4: Factoring
polynomials
 Homework: 5.4 (worksheet)
Warm- up: Lesson 5.4
Alg. 2B
Simplify.
 a b 
1.  0 3 
a b 
12
2
2.
3.
4.
Synthetic division
Factor the following
5. 16 x  8 x
3
2
2
x
 18 x  81
6.
7.
49w 2  100