Name _______________________________
Section ______________
Section 6.6: The Fundamental Theorem of Algebra
Date ________________
1. Write the simplest polynomial function with the given zeros.
!
a. 2, 1, !
b. −4, −1, 2 c. 2 2, 5, −3
d. −2! and ! + 1 e. 1, −1 multiplicity of 3 and 3!
f. 1 − 2 and 2! Gibson 2011-2012
6.6 Homework – page 1
2. Solve each equation by finding all roots. Show the fully factored form, then list all roots.
a. x 3 ! 4x 2 + x +14 = 8
b. x 3 ! 2x 2 ! 2x ! 3 = 0
c. x 4 !13x 3 + 55x 2 ! 91x = 0
d. x 4 + x 2 !12 = 0
Gibson 2011-2012
6.6 Homework – page 2
3. The volume of a rectangular prism is 105 cubic units. Find the dimensions of the prism.
4. The volume of a pyramid-shaped tent with a square base can be represented by the
1
V x = x 3 ! 2x 2
3
function
, where x is the length of the base in meters.
()
a. The volume of the tent is 81 m3. Write a polynomial equation with integer coefficients
that can be solved to find the length of the base.
b. Find the length of the base.
c. What can you say about the other roots of the polynomial equation? Why?
Gibson 2011-2012
6.6 Homework – page 3
5. What is the least degree of a polynomial equation that has 3! as a root with a multiplicity of
3? Explain.
6. Use the graph to estimate the roots of ! = 3!3 − 2!2 − 15! + 10. Then find the exact roots.
(Hint: Factor by grouping.)
7. What is the multiplicity of the root -1 in the equation y = x 4 ! 2x 3 ! 3x 2 + 4x + 4 ?
Gibson 2011-2012
6.6 Homework – page 4
8. Tell whether the statement is sometimes, always, or never true. If it is sometimes true, give
examples to support your answer.
a. A cubic polynomial has no real zeros.
b. A quartic polynomial has an odd number of real zeros.
c. There are infinitely many polynomials with zeros a, b, and c.
d. The multiplicity of a root is equal to the degree of a polynomial.
9. Which polynomial function has zeros 0, !, and − ! ?
10. Which polynomial function has zeros 1 + 3 and 1 − 3?
Gibson 2011-2012
6.6 Homework – page 5
11. A polynomial function has zeros 3 − 2, 4, and 6!. What is the minimum degree of the
polynomial?
12. Use synthetic substitution to evaluate ! ! = ! ! − 3!2 + 9! − 27 for ! = 3! and ! = − 3. Is
either ! = 3! or ! = − 3 a zero of !?
Gibson 2011-2012
6.6 Homework – page 6