Five-Minute Check (over Lesson 5–5)
CCSS
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 5–6
Factor 8c3 – g3. If the polynomial is not factorable,
write prime.
A. (2c)(4c2 + cg + g2)
B. (2c – g)(4c2 + 2cg + g2)
C. (c – g)(2c + g + g2)
D. prime
Over Lesson 5–6
Factor 12az – 6bz – 6cz + 10ax – 5bx – 5cx. If the
polynomial is not factorable, write prime.
A. (2a – 3z)(5x – c)
B. (2a – 6z)(2a + b + c)
C. (5x + 6z)(2a – b – c)
D. prime
Over Lesson 5–6
Factor 8x3m2 – 8x3n2 + y3m2 – y3n2. If the polynomial
is not factorable, write prime.
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
Over Lesson 5–6
Solve 16d4 – 48d2 + 32 = 0.
A.
B.
C.
D.
Over Lesson 5–6
Solve k3 + 64 = 0.
A. 16
B. 8
C. –2
D. –4
Over Lesson 5–6
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.
A. 2 ft
B. 3 ft
C. 4 ft
D. 6 ft
Content Standards
A.APR.2 Know and apply the Remainder
Theorem: For a polynomial p(x) and a number a,
the remainder on division by x – a is p(a), so
p(a) = 0 if and only if (x – a) is a factor of p(x).
F.IF.7.c Graph polynomial functions, identifying
zeros when suitable factorizations are available,
and showing end behavior.
Mathematical Practices
7 Look for and make use of structure.
You used the Distributive Property and
factoring to simplify algebraic expressions.
• Evaluate functions by using synthetic
substitution.
• Determine whether a binomial is a factor of a
polynomial by using synthetic substitution.
• 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.
Synthetic Substitution
Method 2 Direct Substitution
Replace x with 6.
Original function
Replace x with 6.
Simplify.
Answer: By using direct substitution, f(6) = 2453.
If f(x) = 2x3 – 3x2 + 7, find f(3).
A. 20
B. 34
C. 88
D. 142
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.
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?
A. 616,230 students
B. 638,680 students
C. 646,720 students
D. 659,910 students
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
–18
18
0
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)]. 
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)
B. yes; (x + 5)
C. yes; (x + 2)(x + 3)
D. x + 2 is not a factor.