Download x -3 - Standards Aligned System

Chapter 6
Exponents and Polynomials
What You’ll Learn:
• Exponents
• Basic Operations of Nomials
In Class Assignment
• Page 389 # 1-32
Integer Exponents
• Zero Exponents
– Any nonzero number raised to the zero power is 1
• Negative Exponents
– A nonzero number raised to a negative exponent
is equal to 1 divided by that number raised to the
opposite (positive) exponent.
Integer Exponents
55
Power
54
53
52
51
Value
Power
Value
50
5-1
5-2
5-3
5-4
5-5
Ex. 1) Zero and Negative Exponents
Simplify.
A. 2 -3
B. (-3) -4
C. -34
Ex.2 Evaluating Expressions
Evaluate.
A. x-1 for x = -2
B. a0 b-3 for a=8 , b = -2
Ex. 3 Simplify Expressions
Simplify.
A. 3y-2
B. -4
k-4
C. x-3
a 0 y5
Homework
• Practice 6-1 (page395)
– #’s 2-22 even
Rational Exponents
• The radical symbol is used to identify roots
• The index is the small number to the left that
tells which root to take
• Roots of 2,3,4,5,ect…
Writing Roots with Exponents
•
b = bk
Ex. 1) Writing Roots with Exponents
Simplify (w/ and w/out calc)
• 125 1/3
Ex.2 Simplify Expressions
3
• x9 y3
Homework
• Practice 6-2 (page401)
– #’s 2-9;23-30
Polynomials
• Monomial (Term) is a number, a variable, or a
product of numbers and variables with whole
number exponents.
 Ex.
• Polynomial is a monomial or a sum or
difference of monomials
 Ex.
Finding the Degree of Nomials
• Monomial
– Add the exponents of variables
• Ex. -2a2 b4
>>>
• Polynomial is a monomial or a sum or
difference of monomials
– Find the use the degree of the term with the
greatest degree
 Ex. 4x – 18x5
Finding the Degree of Nomials
• Find the degree
a. 4x
b. 2c3
c. X3y2 + x2y3 – x4 + 2
Writing Polynomials in Standard Form
• Standard form of a polynomial
– Polynomials are written with terms arranged in
descending order (greatest degree downward)
Polynomials can also be classified based on their
degree or by how many terms it contains
By degree
• 0 - Constant
• 1- Linear
• 2- Quadratic
• 3- Cubic
• 4- Quartic
• 5-Quintic
• 6 or more- 6th, 7th ,
8th degree…..
By terms
• 1 – Monomial
• 2 – Biomial
• 3 – Trinomial
• 4 or more – Polynomial
Classifying Polynomials
• Classify
o 5x – 6
o y2 + y + 4
o 6x5 + 9x4 – x + 3
Homework
• Practice 6-3 (page 409)
#’s 1-3;4-24 (even)
Algebraic Expression
• In an algebraic expression, a positive or
negative sign is part of the term that follows
it: the term owns the sign that comes before
it.
– An additions sign is understood in front of a
negative sign
Like Terms
• Like Terms have the same variable or variables
raised to the same power.
– In other word in order to be like, Terms can have
different first names (numbers), but must have
the same last name (letter & power)
Examples of Like Terms
•
•
•
•
4x and 9x. Both have x
7xy and 8 xy. Both have xy
2y and 8y . Both have y
Can you think of other like Terms
2
2
2
Non Example of Like Terms
• 4 and 6y. Do not have common factor
• 3x and 3y. Have common factor by not
variable
• 5y and 6y. Do not have the same power
• Can you think of other examples???
2
Combining (add/subtract) Like Terms
• Simplify: 4x + 6y – 3x – 4y
• Group like terms (x & y terms) [the sign travels
with the terms]
• (4x + - 3x) + (6y + - 4y)
• Combine like terms
– x + 2y
Adding Polynomials
• Add 3x + 4xy – 2y + 3 and x y + 3y - 4
2
2
3
2
3
– Group like terms
– Combine like terms
• You can also align similar terms vertically and
then add
Practice Adding Polynomials
Simplify Horizontally
•
3xy + 4x – 2y + 3 and x y + 3y - 4
Alg Bk. p. 149
Simplify Vertically
• 3x y + 4x – 2y + 3
• xy+
3y - 4
_________________
Subtract Polynomials
• Subtract –a
2
2
- 5ab + 4b - 2 from 3a – 2ab – 2b
• Add the opposite of the second polynomial
2
– Change sign of terms
• Group like terms
• Combine like terms
2
Simplify :
• ( –a - 5ab + 4b) – (3a – 2ab – 2b)
• (3x2 - 2x + 8 ) - (x2 - 4)
• (4b5 + 8b) + (3b5 + 6b - 7b5 + b)
Homework
• Practice 6-4 (page 417)
o Day 1
o #’s 1-14
o Day 2
o #’s 16-32
Exponential Notation
• An exponent is a number
that represents how
many times the base is
used as a factor. For
example, the number 8
with an exponent of 4 is
equal to 8 x 8 x 8 x 8.
• Base
• Exponent
Multiplying Monomials
• Rule of Exponents for products of powers
– To multiply two powers with the same base, you
add the exponents
• (x 4 ) (x 2 ) = x 6
• (2x 3 ) (4x 3 ) = 8x
6
Multiply Monomials
•
•
•
•
•
•
(x ) (x )
(y ) (y) (y )
2s(5s)
(4y z)(2yz )
(5x y)(3x y )
(-3s) (7s )
2
5
3
2
6
6
5
2
2
2
Simplify Products of Monomials
3
• (3x y )(-2x y) + (8x y ) (x y )
4
6
2
2
3
5
Multiplying Polynomials
• By using the distributive property and the
rules of exponents, any polynomial can be
multiplied
• Two methods for doing so:
– Horizontal method
– Vertical method
Multiply x (x+3)
Horizontal Method
– x (x + 3)
Vertical Method
• x+3
2
• x + 3x
• x
____________
2
• x + 3x
Multiply -2x(4x - 3x + 5)
Horizontal Method
Vertical Method
Multiply 5xy (3x - 4xy + y )
2
Horizontal Method
2
Vertical Method
2
Multiply (3x – 2) (2x - 5x – 4)
2
2
2x -5x - 4
3x – 2
_____________
• Multiply by the 3x first
• Multiply by the -2
• Add/combine like terms
3
Answer: 6x - 19x -22x + 8
2
• It is helpful to rearrange the terms in either
ascending or descending order
– Descending order
3
2
• x + 2x - 4x + 2
– Ascending order
• 2 – 4x + 2x + x
2
3
Multiplying Binomials
• When multiplying binomials the product
results in a trinomial
– (a + b) (c + d) = ac + ad + bc + bd
• In to multiply we must use the distributive
property
• We call this Method of multiplication FOIL
FOIL
(a + b ) (c + d)
• F – Firsts
– (a + b ) (c + d) = ac
• O – Outers
– (a + b ) (c + d) = ad
• I – Inners
– (a + b ) (c + d) = bc
• L – Lasts
– (a + b ) (c + d) = bd
Write the product of (2x + 5) (3x- 4)
• FOIL
– Firsts: (2x)(3x) = 6x 2
– Outers: (2x)(-4) = -8x
– Inners: (5)(3x) = 15x
– Lasts: (5)(-4) = -20
2
• 6x – 8x +15x – 20
2
• 6x + 7x -20
Practice Foil
•
•
•
•
•
•
•
(x+1)(x+8)
(y+2)(y+5)
(t-5)(t-3)
(u-2)(u-1)
(s-9)(s+9)
(8k-1)(k+3)
2
(2n+4)
Homework
• Practice 6-5 (page 427)
o Day 1
o #’s 2-24 even
o Day 2
o #’s 1-23 odd
Special Product Binomials
• Perfect-Square Trinomial - A trinomial that is
the result of squaring a binomial
 (a + b ) 2
 (a + b) (a – b)
Ex 1) Find the Product (a + b ) 2
• Multiply
o (x + 4) 2
o (3x + 2y) 2
o (4 + s2) 2
Ex 2) Find the Product (a - b ) 2
• Multiply
o (x - 5) 2
o (6a - 1) 2
o (3 – x2 ) 2
Ex 2) Find the Product (a + b ) (a – b )
• Multiply
o (x + 6 ) (x – 6 )
o (x2 + 2y ) (x2– 2y )
Homework
• Pr 6-6 (p. 437)
o Day 1 # 2- 18 even
o Day 2 # 21-37 odd
Chapter Review
• Text p.. 442 – 445
– Selected problems