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
Product of Powers
a m  a n  a m n
Power of a Power
a 
Power of a Product
 ab 
Zero Exponent
m
n
m
 a m n
 a mb m
a0  1, a  0
Ex. 1: Evaluate the expressions:
a.
 3  5 
2
3
b.

b.
 8 8

13 0
322   9
You Try: Evaluate the expressions:
a.
4 
2
3
3
*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
n
a
Power of a Quotient
am
a

,b  0
 
bm
b
m
Ex. 4: Evaluate each expression.
 53 
a.  7 
5 
1
6 104
b.
9 107
You Try:
 14 
a.  2 
3 
2
Algebra 2 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. 2: Write the answer in scientific notation:
 6.3 10 8.9 10 
5
12
You Try: A Federal Reserve gold bar weighs 400 ounces. What is the weight of 250,000,000
of the gold bars?
Simplify the expressions.
Ex. 3: x 6 x 5 x 3
You Try:
 7 y z  y z 
2 5
4 1
Ex. 5: Simplify the expression.
 c 
b.  4 
d 
a. w5 w8 w6
2
c.
a 3b 2
a5
You Try: Simplify the expression.
 s3 
a.  4 
t 
2
 x 4 y 2 
b.  3 6 
 x y 
3
3
 a 2 b 1 
You Try: What is the simplified form of  3 2  ?
 2a b 
b9
A. 8a 3b 6
B.
6a 3
C.
1
8a 3b 6
D.
1
8a 3b 9
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
Algebra 2 Ch 5A Notes Page 2
5-2
Evaluate and Graph Polynomial Functions
Name _______________
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 are all whole numbers.
A polynomial function is in standard form if its terms are written in descending order.
*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.
Ex. 2: Use synthetic substitution to evaluate f ( x)  2 x 4  3x3  6 x 2  3 when x  2 .
Algebra 2 Ch 5A Notes Page 3
*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
Graphing Polynomial Functions: To graph a polynomial function, first plot points to determine
the graph’s middle portion. Then use what you know about end behavior to sketch the ends of
the graph.
Ex.4: Graph
a. f ( x)  x3  x 2  4 x  4
Algebra 2 Ch 5A Notes Page 4
b. f ( x)   x3  10 x 2  25x
5-3 Add, Subtract, and Multiply Polynomials
Name____________________
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)
Algebra 2 Ch 5A Notes Page 5
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.
Algebra 2 Ch 5A Notes Page 6