Polynomials and Polynomial Functions Chapter 5
Prerequisite Skills
VOCABULARY CHECK
Copy and complete the statement 1. The zeros of the function graphed are ? . ANSWER – 3 and 1
2. The maximum value of the function graphed is ? . ANSWER 4
Prerequisite Skills
VOCABULARY CHECK
3. The standard form of a quadratic equation in one variable is ? where a = 0. ANSWER y = ax2 + bx + c
Prerequisite Skills
SKILLS CHECK
Graph the function. Label the vertex and the axis of symmetry. 4. y = – 2(x – 1)2 + 4 ANSWER 5. y = 3(x – 2)(x + 3) ANSWER
6. y = – x2 – 4x + 4 ANSWER
Prerequisite Skills
SKILLS CHECK
Factor the expression. 7. x2 + 9x + 20
ANSWER (x + 4)(x + 5)
8. 2x2 + 5x – 3
ANSWER (2x + 1)(x – 3)
9. 9x2 – 64
ANSWER (3x – 8)(3x + 8)
Prerequisite Skills
SKILLS CHECK
Solve the equation. 10. 2x2 + x + 6 = 0
ANSWER
– 1 + i √47 4
11. 10x2 + 13x = 3
12. x2 + 6x + 2 = 20
ANSWER – 1.5, 0.2 ANSWER – 3 + 3 √3
Properties of Exponents Ch. 5, Sec. 1
Objectives: 1) Use properties of exponents to evaluate and simplify exponential expressions. 2) Use powers as models.
a
exponent
n
a is used as a factor n times read >> a to the nth power
base Try this: a.
2 •2 4
5
b.
3 3 3
2
Properties of Exponents Let a and b be real numbers and let m and n be integers
a •a = a m
n
m+ n
m•n
(a ) = a m m m (ab) = a b m n
a
−n
1 = n , a≠0 a
a = 1, a ≠ 0 0
m
a m− n =a , a≠0 n a
FG a IJ H bK
m
m
a = m , b≠0 b
Example 1 5 ( − 3 )( − 3 ) a. b.
( −2)
3 3
2
c. (3 xy ) d.
4
4 −3 • 4 3 −2 2
e.
(3 )
f.
1 x • 5 x
g.
62 • 6 65
3
i.
FG 2 IJ H 3K
−2
EXAMPLE 1
Use scientific notation in real life
Locusts A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 square kilometers. About how many locusts are in such a swarm?
EXAMPLE 3 a.
b.
c.
Simplify expressions
b–4b6b7 r–2 s3
= b–4 + 6 + 7 = b9 –3
Product of powers property
– 2 )–3 ( r = ( s3 )–3
Power of a quotient property
6 r = –9 s
Power of a power property
= r6s9
Negative exponent property
16m4n –5 4n – 5 – (–5) = 8m 2n–5 = 8m4n0= 8m4
Quotient of powers property Zero exponent property
EXAMPLE 4
Standardized Test Practice
EXAMPLE 5
Compare real-life volumes
Astronomy Betelgeuse is one of the stars found in the constellation Orion. Its radius is about 1500 times the radius of the sun. How many times as great as the sun’s volume is Betelgeuse’s volume? Be·tel·geuse [büt'l jz, büt'l jz] noun
Alpha Orionis
red star in Orion: a bright red variable supergiant star that is the second brightest star in the constellation Orion and the twelfth brightest in the night sky Microsoft® Encarta® 2006. © 1993-2005 Microsoft Corporation. All rights reserved.
EXAMPLE 5
Compare real-life volumes
SOLUTION Let r represent the sun’s radius. Then 1500r represents Betelgeuse’s radius. Betelgeuse’s volume = Sun’s volume
=
4 π (1500r)3 3 4 π r3 3
The volume of a sphere is 4 πr3. 3
4 π 15003r3 3 4 π r3 3
Power of a product property
EXAMPLE 5
Compare real-life volumes = 15003r0 = 15003
Quotient of powers property
1
= 3,375,000,000
ANSWER
Zero exponent property Evaluate power.
Betelgeuse’s volume is about 3.4 billion times as great as the sun’s volume.
GUIDED PRACTICE
for Examples 3, 4, and 5
Simplify the expression. Tell which properties of exponents you used. 5.
x–6x5 x3
SOLUTION x–6x5x3
= x–6x5 + 3
Power of a product property
= x2
Simplify exponents.
GUIDED PRACTICE 6.
for Examples 3, 4, and 5
(7y2z5)(y–4z–1)
SOLUTION (7y2z5)(y–4z–1)
= (7y2z5)(y–4z–1)
Power of a product property
= (7y2 – 4)(z5 +(–1)) Simplify = (7y–2)(z4) = 7z4 y2
Negative exponent property
for Examples 3, 4, and 5
GUIDED PRACTICE 7.
s3
2
t–4
SOLUTION s3
t–4
2
=
s (3)2 (t–4 )2
Power of a product property
=
s6 t–8
Evaluate power.
=
s6t8
Negative exponent property
GUIDED PRACTICE 8.
x4y–2 x3y6
for Examples 3, 4, and 5
3
SOLUTION x4y–2 x3y6
3
(x4)3 (y–2)3 = (x3)3(y6)3 =
x12y–6 x9y18
Power of a powers property
Power of a powers property
= x3y–24
Power of a Quotient property
x3 = 24 y
Negative exponent property
Lesson Close: 1. When multiplying powers, what is the importance having the “same base”? 2. Do the properties of exponents hold for decimals as 5.6 and 3.45?
Assignment: Pg. 333-335 >> 2, 16-48 eoe, 50, 52
