3-7 The Real Zeros of a Polynomial Function
p. 258: 63-67 odd, and Remainder/Factor Worksheet
I. The Remainder Theorem
If f(x) is a polynomial function and if f(x) is divided by x – c, then the remainder is f(c).
II. The Factor Theorem
Let f(x) be a polynomial function.
A. If f(c) = 0, then x – c is a factor of f(x).
B. If x – c is a factor of f(x), then f(c) = 0.
III. The Intermediate Value Theorem
Let f be a continuous function on the closed interval [a,b]. If f(a) and f(b) have opposite signs, then
f contains at least one real zero in the open interval (a,b).
1. Find the remainder when px   x15  3x10  2 is divided by x  1
2. Find the quotient and remainder when the first polynomial is divided by the second.
x 3  2 x  18; x  2
3. Determine whether x –2 or x+2 is a factor of x 5  3 x 2  20
4. When a polynomial p(x) is divided by x+3, the quotient is 2 x 2  3 x  9 and the remainder is -11. Find p(x).
Given a polynomial and one or more of its roots, find the remaining roots.
5. 2 x 3  5 x 2  23x  10  0; root x  2
6. 2 x 4  5 x 3  8x 2  17 x  6; roots x  -1and x  2
Use the Intermediate Value Theorem to show that each function has a zero in the given interval. Approximate
the zero rounded to three decimal places.
7. f ( x)  x 3  3x  1; [0,1]
8. f x   2 x 4  x 3  6 x  3 ; [1,2]
Descartes Rule of Signs
Chapter 3
Polynomials/Descartes Worksheet
I. Descartes Rule of Signs
Let p(x) be a polynomial:
A. The number of positive real zeros equals the # of sign changes in p(x) or the number of sign changes
minus two.
B. The number of negative real zeros equals the # of sign changes in p(-x) or the number of sign changes
minus two.
II. Upper Bound Theorem
If p(x) is divided by x – c and there are no sign changes in the quotient or remainder, then c is upper bound.
III. Lower Bound Theorem
If p(x) is divided by x + c and there are alternating sign changes in the quotient and the remainder, then -c
Is the lower bound.
IV. The Fundamental Theorem of Algebra
A polynomial function of degree n has at most n real zeros.
V. Rational Roots Theorem
If f ( x)  qx n  cx n1  ...  c1 x  p , then factors of 
p
are possible rational zeros of f where the
q
coefficients of f are integers.
Use Descartes Rule of signs to summarize the number of positive, negative, and imaginary roots of each
polynomial equation.
1. 2 x 4  x 3  7 x 2  4 x  1  0
2. x 3  6x 2  1  0
Find the upper and lower bound of each polynomial equation and then state the interval in which the roots lie.
3. 2 x 4  x 3  7 x 2  4 x  1  0
4. 2 x 3  x 2  28x  51  0
Solve each equation, giving all real and imaginary roots. (Remember: Some of these may factor)
5. x 4  2 x 3  2 x 2  6 x  3  0
6. x 3  x 2  x  1  0
7. 3x 4  5 x 2  2  0
8. 3x 3  4 x 2  5 x  2  0
9. x 4  2 x 3  x 2  4 x  2  0
3-6 Polynomial Inequalities
Polynomial and Rational Inequalities Worksheet
I. Solving Polynomial Equations
A. Collect all terms to one side compared to zero
B. Factor and solve for x
C. Perform an interval line test to find the desired values of positive or negative
D. Express the solution set in interval notation
Solve each inequality
1. x  1 x  3  0
2. x 2  3x  0
4. 2 x 2  3x  5
5. x x  5  14
6.
7. x 3  4 x 2
8. x 4  4 x 2  5  0
9. x 3  2 x 2  15x  0
b gb g
( x  2)( x  5) 2
0
10.
( x  1) 2
b g
3. x 2  16  0
bx  2gcx  6x  9h 0
2