• synthetic substitution
• depressed polynomial
Synthetic Substitution
If f(x) = 2x4 – 5x2 + 8x – 7, find f(6).
Method 1 Synthetic Substitution
By the Remainder Theorem, f(6) should be the
remainder when you divide the polynomial by x – 6.
2
2
0
–5
12
72
12
67
–7
Notice that there is no
x3 term. A zero is
402 2460 placed in this position
as a placeholder.
410 2453
8
Answer: The remainder is 2453. Thus, by using
synthetic substitution, f(6) = 2453.
If f(x) = 2x3 – 3x2 + 7, find f(3).
A. 20
B. 34
C. 88
D. 142
A.
B.
C.
D.
A
B
C
D
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?
2011- 1993 = 18
S(18) = 451,760
Answer: In 2011, there will be about 451,760 U.S.
college students studying abroad.
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?
B. 638,680 students
C. 646,720 students
0%
B
D. 659,910 students
A
0%
0%
C
A. 616,230 students
A
B
C
D
0%
D
A.
B.
C.
D.
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
4 –15
3
21
1
7
6
x2 + 7x + 6
–18
18
0
Use the Factor Theorem
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).
Determine whether x + 2 is a factor of
x3 + 8x2 + 17x + 10. If so, find the remaining factors
of the polynomial.
A. yes; (x + 5)(x + 1)
0%
B
D. x + 2 is not a factor.
A
0%
A
B
C
0%
D
D
C. yes; (x + 2)(x + 3)
A.
B.
C.
0%
D.
C
B. yes; (x + 5)