3.6 Homework
Use the Factor Theorem to determine whether x  c is a factor of f  x 
1.
f  x   3x4  6x3  5x  10; x  2
2.
f  x   3x6  82x3  27; x  3
3.
f  x   4 x6  64 x4  x2 15; x  4
4.
f  x   2 x 4  x 3  2 x  1; x 
1
2
5.
a) Tell the maximum number of zeros that each polynomial may have.
b) List the potential rational zeros of each function
c) Then use Descartes rule of signs to determine how many possible positive and negative zeros
each polynomial function may have. Do not attempt to find the zeros.
6.
f  x   4 x7  x3  x2  2
f  x   2 x6  3x2  x  1
7.
f  x   3x 3  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  8x 4  x  2
11. Solve and sketch the equation in the real number system.
x 4  x3  2 x 2  4 x  8  0
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c)
Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
f)
Using part ‘d’ sketch the graph
12. Solve and sketch the equation in the real number system.
3x3  x 2  8 x  4  0
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c)
Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
f)
Using part ‘d’ sketch the graph
13. Solve and sketch the equation in the real number system.
3x3  x 2  27 x  9  0
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c)
Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
f)
Using part ‘d’ sketch the graph
14. Solve and sketch the equation in the real number system.
x 4  3x3  3x 2  7 x  6  0
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c)
Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
f)
Using part ‘d’ sketch the graph
Use the Intermediate Value Theorem to show that each polynomial function
has a zero in the given interval.
15. f  x   8x4  2 x2  5x  1;
0,1
16. f  x   2 x3  6 x2  8x  2;  5,  4
17. Find k such that f  x   x3  kx2  kx  2 has the
18. Find k such that f  x   x4  kx3  kx2  1 has the
factor x  2
factor x  2
19. What is the remainder when
f  x   2 x20  8x10  x  2 is divided by x 1 ?
20. One solution of the equation
x3  4 x 2  7 x  10  0 is -2.
Find the sum of the remaining solutions.
(Think simple)