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Precalculus Homework:
3.6 Real Zeros of a Polynomial
Name: ____________________________
Date: _________________ Block: ______
Use the Factor Theorem to determine whether ( x − c ) is a factor of f ( x )
1.
f ( x ) = 3 x 4 − 6 x3 − 5 x + 10; x − 2
2.
f ( x ) =3 x 6 + 82 x3 + 27; x + 3
3.
f ( x ) = 4 x 6 − 64 x 4 + x 2 − 15; x + 4
4.
f ( x ) = 2 x 4 − x 3 + 2 x − 1; x −
a) Tell the maximum number of zeros that each polynomial may have.
b) List the potential rational zeros of each function. (You do not need to find the zeros.)
5.
−4 x 7 + x3 − x 2 + 2
f ( x) =
6.
f ( x )= 2 x 6 − 3 x 2 − x + 1
7.
f ( x )= 3 x3 − 2 x 2 + x + 2
8.
f ( x) =
− x4 + x2 −1
9.
f ( x ) = 2 x3 − x 2 + 2 x − 1
10.
f ( x )= 4 x 4 − 8 x 4 − x + 2
1
2
11. Solve and sketch the equation in the real number system.
a)
x 4 − x3 + 2 x 2 − 4 x − 8 =
0
How many possible real roots are there?
_______, _______, or _______
b)
Use Rational Zero Theorem to find the possible rational zeros
c)
Use synthetic or long division to test possible rational zeros
d)
Write that zero as a factor, along with the “depressed polynomial”. Factor the depressed polynomial.
e)
Solve, based on factors, over real numbers.
f)
Using part e, along with your knowledge of end behavior, sketch the graph
12. Solve and sketch the equation in the real number system.
a)
3x3 + x 2 − 8 x + 4 =
0
How many possible real roots are there?
_______, or _______
b)
Use Rational Zero Theorem to find the possible rational zeros
c)
Use synthetic or long division to test possible rational zeros
d)
Write that zero as a factor, along with the “depressed polynomial”. Factor the depressed polynomial.
e)
Solve, based on factors, over real numbers.
f)
Using part e, along with your knowledge of end behavior, sketch the graph
13. Solve and sketch the equation in the real number system.
a)
3 x 3 − x 2 − 27 x + 9 =
0
How many possible real roots are there?
_______, or _______
b)
Use Rational Zero Theorem to find the possible rational zeros
c)
Use synthetic or long division to test possible rational zeros
d)
Write that zero as a factor, along with the “depressed polynomial”. Factor the depressed polynomial.
e)
Solve, based on factors, over real numbers.
f)
Using part e, along with your knowledge of end behavior, sketch the graph
14. Solve and sketch the equation in the real number system.
a)
x 4 + 3x3 − 3x 2 − 7 x + 6 =
0
How many possible real roots are there?
_______, _______, or _______
b)
Use Rational Zero Theorem to find the possible rational zeros
c)
Use synthetic or long division to test possible rational zeros
d)
Write that zero as a factor, along with the “depressed polynomial”. Factor the depressed polynomial.
e)
Solve, based on factors, over real numbers.
f)
Using part e, along with your knowledge of end behavior, sketch the graph
Use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval.
15. f ( x ) = 8 x 4 − 2 x 2 + 5 x − 1;
[0,1]
16. f ( x )= 2 x 3 + 6 x 2 − 8 x + 2; [ −5, − 4]
17. Find k such that f ( x ) = x 3 − kx 2 + kx + 2 has the
18. Find k such that f ( x ) =x 4 − kx 3 + kx 2 + 1 has the
factor x − 2
factor x + 2
19. What is the remainder when
f ( x=
) 2 x 20 − 8 x10 + x − 2 is divided by x − 1 ?
20. One solution of the equation
(Think simple)
Find the sum of the remaining solutions.
x3 − 4 x 2 − 7 x + 10 =
0 is -2.