View
223
Download
0
Category
Preview:
Citation preview
Five-Minute Check (over Lesson 6–5)
Then/Now
New Vocabulary
Key Concept: Remainder Theorem
Example 1:Synthetic Substitution
Example 2:Real-World Example: Find Function Values
Key Concept: Factor Theorem
Example 3:Use the Factor Theorem
Over Lesson 6–6
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. (2c)(4c2 + cg + g2)
B. (2c – g)(4c2 + 2cg + g2)
C. (c – g)(2c + g + g2)
D. prime
Factor 8c3 – g3. If the polynomial is not factorable, write prime.
Over Lesson 6–6
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. (2a – 3z)(5x – c)
B. (2a – 6z)(2a + b + c)
C. (5x + 6z)(2a – b – c)
D. prime
Factor 12az – 6bz – 6cz + 10ax – 5bx – 5cx. If the polynomial is not factorable, write prime.
Over Lesson 6–6
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. (8x + y)(m + n)(m – n)
B. (4x + y)(2x – y2)(m + n)(m – n)
C. (2x + y)(4x2 – 2xy + y2)(m + n)(m – n)
D. prime
Factor 8x3m2 – 8x3n2 + y3m2 – y3n2. If the polynomial is not factorable, write prime.
Over Lesson 6–6
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
Solve 16d4 – 48d2 + 32 = 0.
A.
B.
C.
D.
Over Lesson 6–6
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 16
B. 8
C. –2
D. –4
Solve k3 + 64 = 0.
Over Lesson 6–6
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 2 ft
B. 3 ft
C. 4 ft
D. 6 ft
The width of a box is 3 feet less than the length. The height is 4 feet less than the length. The volume of the box is 36 cubic feet. Find the length of the box.
You used the Distributive Property and factoring to simplify algebraic expressions. (Lesson 5–3)
• Evaluate functions by using synthetic substitution.
• Determine whether a binomial is a factor of a polynomial by using synthetic substitution.
Synthetic Substitution
If f(x) = 2x4 – 5x2 + 8x – 7, find f(6).
Method 1 Synthetic Substitution
Answer: The remainder is 2453. Thus, by using synthetic substitution, f(6) = 2453.
By the Remainder Theorem, f(6) should be the remainder when you divide the polynomial by x – 6.
2 12 67 410 2453
Notice that there is no x3 term. A zero is placed in this position as a placeholder.
2 0 –5 8 –7
12 72 402 2460
Synthetic Substitution
Method 2 Direct Substitution
Replace x with 6.
Answer: By using direct substitution, f(6) = 2453.
Original function
Replace x with 6.
Simplify.
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 20
B. 34
C. 88
D. 142
If f(x) = 2x3 – 3x2 + 7, find f(3).
Find Function Values
COLLEGE The number of college students from the United States who study abroad can be modeled by the function S(x) = 0.02x
4 – 0.52x
3 + 4.03x
2 + 0.09x + 77.54, where x is the number of years since 1993 and S(x) is the number of students in thousands. How many U.S. college students will study abroad in 2011?
Answer: In 2011, there will be about 451,760 U.S. college students studying abroad.
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 616,230 students
B. 638,680 students
C. 646,720 students
D. 659,910 students
HIGH SCHOOL The number of high school students in the United States who hosted foreign exchange students can be modeled by the function F(x) = 0.02x
4 – 0.05x
3 + 0.04x
2 – 0.02x, where x is the number of years since 1999 and F(x) is the number of students in thousands. How many U.S. students will host foreign exchange students in 2013?
Use the Factor Theorem
Determine whether x – 3 is a factor of x3 + 4x2 – 15x – 18. Then find the remaining factors of the polynomial.
The binomial x – 3 is a factor of the polynomial if 3 is a zero of the related polynomial function. Use the factor theorem and synthetic division.
1 7 6 0
1 4 –15 –18
3 21 18
Use the Factor Theorem
Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial x3 + 4x2 – 15x –18 can be factored as (x – 3)(x2 + 7x + 6). The polynomial x2 + 7x + 6 is the depressed polynomial. Check to see if this polynomial can be factored.
x2 + 7x + 6 = (x + 6)(x + 1) Factor the trinomial.
Answer: So, x3 + 4x2 – 15x – 18 = (x – 3)(x + 6)(x + 1).
Use the Factor Theorem
Check You can see that the graph of the related function f(x) = x3 + 4x2 – 15x – 18 crosses the x-axis at 3, –6, and –1. Thus, f(x) = (x – 3)[x – (–6)][x – (–1)].
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. yes; (x + 5)(x + 1)
B. yes; (x + 5)
C. yes; (x + 2)(x + 3)
D. x + 2 is not a factor.
Determine whether x + 2 is a factor of x3 + 8x2 + 17x + 10. If so, find the remaining factors of the polynomial.
Recommended