Download 5-1A Use Properties of Exponents

5-1 Use Properties of Exponents
Name _______________
Objective: To simplify expressions involving powers
Algebra 2 Standard 7.0
*Properties of Exponents: Let a and b be real numbers and let m and n be integers.
Property Name
Definition
Example
a m  a n  a m n
(add them if they have the same base)
Product of Powers
Power of a Power
a 
Power of a Product
 ab 
Zero Exponent
m
n
m
 a m n
 a mb m
(an exponent applies to every factor inside)
a 0  1, when a  0
*An exponent applies only to what is immediately in front of it.
(See the difference!) 32  9 and (  3)2  9
Ex. 1: Evaluate the expressions:
a.
 3  5 
2
3
b.
 3
22

13 0
  9
You Try: Evaluate the expressions:
a.
3  4 
2
b. 82  8 
3
2
*Properties of Exponents: Let a and b be real numbers and let m and n be integers.
Property Name
Definition
Example
1
,a  0
am
Negative Exponent
am 
Quotient of Powers
am
 a mn , a  0
an
Power of a Quotient
am
a

,b  0
 
bm
b
m
*Never leave the negative exponents or (parenthesis) in the final answer!
Ex. 2: Evaluate each expression.
 53 
a.  7 
5 
1
b.
6 104
9 107
You Try:
 14 
a.  2 
3 
2
Alg 2 Adv Ch 5A Notes Page 1
 2 
b.  2 
3 
3
*Scientific Notation: A number is expressed in scientific notation
if it is in the form c  10n where 1 < c < 10 and n is an integer.
Ex. 3: Write the answer in scientific notation.
a.
 6 10 8 10 
b.
21105
c.
0.003 104
d.
1.08 1025
4.32 1023
e.
880 107
11103
5
12
Simplify the expressions.
Ex. 4: x 6 x 5 x 3
You Try: w5 w8 w6
Ex. 5: Simplify the expression.
a.
 7 y z  y z 
4 1
2 5
 c 
b.  4 
d 
2
a 3b 2
c.
a5
You Try: Simplify the expression.
a.
 s3 
 4 
t 
2
 x 4 y 2 
b.  3 6 
 x y 
3
 a 2 b 1 
c.  3 2 
 2a b 
3
Ex. 6: The radius of Jupiter is about 11 times greater than the radius of Earth. How many times
as great as Earth’s volume is Jupiter’s volume?
4
Hint: volume of a sphere is V   r 3
3
Alg 2 Adv Ch 5A Notes Page 2
5-2
Evaluate and Graph Polynomial Functions
Objective: To evaluate and graph other polynomial functions.
*Polynomial Function:
f  x   an xn  an1xn1    a1x  a0
where an  0 and an is the leading coefficient, n is the degree, and a0 is the constant term.
The exponents of polynomial functions are all whole numbers.
A polynomial function is in standard form if its terms are written in descending order.
<New Vocabulary>
1. polynomial:
2. Terms are separated by (+) or (-) sign.
3. degree
4. leading coefficient
5. whole numbers:
*Classification of Polynomial functions
Example
Degree
0
Name
Constant function
1
Linear function
2
Quadratic function
3
Cubic function
4
Quartic funciton
Ex. 1: Decide whether the function is a polynomial function. If so, write it in standard form and
state the degree, type and leading coefficient.
a. f
 x   6 x1/ 2  5 x
c. f  x   x 
3
4
x 1
5
b. g  x   8 x  4 x  10  x
5
2
d.
2
4
h  x   3x4  9 x2  4  x4
*Synthetic Substitution: An alternate method to evaluate a polynomial function
using fewer operations than direct substitution.
*How to set up for synthetic substitution
1. Write in descending order.
2. If there is a missing term, make sure to put zero as a place holder.
Alg 2 Adv Ch 5A Notes Page 3
Ex. 2: Use synthetic substitution to evaluate the given function when x  2 .
a. f ( x)  2 x 4  3x3  6 x 2  3
b. g ( x)  x3  5 x 2  6 x  1
*How to sketch polynomial functions:
The End Behavior of a function’s graph is the behavior of the graph as x approaches positive
infinity    or negative infinity    . For the graph of the polynomial function, the end
behavior is determined by the function’s degree and the sign of the leading coefficient.
Even-degree
Odd-degree
(+)
leading
coefficient
(-)
leading
coefficient
Ex. 3: What is true about the degree and leading coefficient of the polynomial whose graph is
shown at the right?
A.
B.
C.
D.
Degree is odd, leading coefficient is positive
Degree is odd, leading coefficient is negative
Degree is even, leading coefficient is positive
Degree is even, leading coefficient is negative
You Try: Page 339 # 8
Alg 2 Adv Ch 5A Notes Page 4
*Graphing Polynomial Functions: To graph a polynomial function, first plot points to determine
the graph’s middle portion. Then use what you know about the end behavior to sketch the ends
of the graph.
Ex.4: Graph:
a. f ( x)  x3  x 2  4 x  4
b. g ( x)   x3  10 x 2  25x
c. h( x)   x 4  5x 2  6
You Try: Graph.
Alg 2 Adv Ch 5A Notes Page 5
a. f ( x)  x 4  16
b. y   x3
5-3 Add, Subtract, and Multiply Polynomials
Objective: To add, subtract and multiply polynomials
Algebra 2 Standard 3.0
*To add or subtract polynomials, add or subtract the coefficients of like terms.
Ex. 1: Add the polynomials vertically and horizontally.
4 x3  4 x 2  3x  10
a.
b. (3y3 – 2y2 – 7y) + (-4y2 + 2y – 5)
3
2
 5 x  2 x  4 x  4


Ex. 2: Subtract the polynomials vertically and horizontally.
3x3  4 x 2  7 x  12
a.
b. (3y2 – 4y + 7) – (6y2 – 6y – 13)
 4 x3  6 x 2  9 x  3


You Try: Find the sum or difference using either format.
a. (t2 – 6t + 2) + (5t2 – t – 8)
b. (8d – 3 + 9d3) – (d3 – 13d2 – 4)
*To multiply two polynomials, you multiply each term of the first polynomial by each term of
the second polynomial.
Ex. 3: Multiply the polynomials vertically and horizontally.
a.
 3x

2
 3x  5

 2 x  3
b. (x – 5)(x2 – 2x + 3)
*Multiply three binomials: multiply the first two binomials and then multiply by the third binomial.
Ex. 4: (x – 1)(x – 1)(x + 2)
Alg 2 Adv Ch 5A Notes Page 6
You Try: Find the product.
a. ( x + 2)(3x2 – x – 5)
b. (y – 5)(y + 2)(y + 6)
Example
*Special Product Patterns
Sum and Difference
(a + b) (a – b)= a2 – b2
Square of a binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Cube of a binomial
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3
Ex. 5: Find the product of the binomials.
a. (5y – 3) (5y + 3)
b. (mn – 6)3
You Try: Find the product.
a. (4a + 7)2
b. (2x + 3)3
Ex. 6: New highway markers are placed every (6x – 6) feet along a stretch of highway. The
total number of markers is represented by x2 – 3x + 1. Write a model for the distance along the
highway where the markers are placed.
Alg 2 Adv Ch 5A Notes Page 7
5-4 Factor and Solve Polynomial Equations
Objective: To factor other polynominal equations.
Algebra 2 Standard 4.0
*Factoring polynomials: A factorable polynomial with integer coefficients is factored completely
if it is written as a product of unfactorable polynominals with integer coefficients.
Factored completely:
Not factored completely:
Ex. 1: Factor the polynomial completely.
a. y3 – 4y2 – 12y
b. 3x3 + 30x2 + 75x
c. 5g5 – 80g3
You Try: Factor the polynomial completely.
a. x3 – 7x2 + 10x
b. 3y5 – 75y3
*Special Factoring Patterns:
Ex.
Sum of Two Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
Ex.
Difference of Two Cubes
a3 – b3 = (a – b)(a2 + ab + b2)
Ex. 2: Factor the polynomial completely.
a. 27x3 + 125
b. -2d5 + 250d2
You Try: Factor the polynomial completely.
a. 8b5 + 343b2
b. w3 - 27
*Factoring by Grouping: For some polynomials, you can factor by grouping pairs of terms
that have a common monomial factor. The pattern is shown below:
ra  rb  sa  sb  r  a  b   s  a  b 
  a  b  r  s 
Alg 2 Adv Ch 5A Notes Page 8
Ex. 3: Factor the polynomial completely.
You Try: Factor the polynomial completely.
27r3 + 45r2 – 3r – 5
x3 + 7x2 – 9x – 63
*Quadratic Form: An expression in the form au2 + bu + c, where u is any expression in x,
is said to be in Quadratic Form.
Ex. 4: Factor the polynomial completely.
a. 10x4 – 10
b. 3m12 + 54m7 + 51m2
You Try: Factor the polynomial completely.
a. 16g4 – 625
b. 4h6 – 20h4 + 24h2
*Solving polynomial equations: To solve polynomial equations by factoring,
factor the polynomial and use the zero product property.
What are the real number solutions of each equation?
Ex. 5: 4x5 – 40x3 + 36x = 0
Ex. 6: 2x5 = 12x3 – 16x
You Try:
a. 2x5 + 24x = 14x3
Alg 2 Adv Ch 5A Notes Page 9
b. 6x3 + x2 = 2x
Ex. 7: -27x3 + 15x2 = -6x4
c. 17x3 + 6x2 = 3x4
* Real Zeros of Polynomial Functions: If f is a polynomial function and a is a real number,
the following statements are equivalent:
1.
x=a is a zero of the function f(x).
2.
x=a is a solution to f(x)=0.
3. (x–a) is a factor of f(x).
4. (a, 0) is an x-intercept of the graph of f(x).
Alg 2 Adv Ch 5A Notes Page 10