Wednesday, December 5, 2012
These problems are from page 388 #s 31-51 every other odd. Factor each polynomial as the
product of linear factors. Check your answers in the back of your book.
31. P  x   x 3  5x2  x  5
35. P  x   x 3  x2  18
39. P  x   x 4  4 x 3  x2  16x  20
43. P  x   x 4  3x 3  x2  13x  10
47. P  x   x 6  12x 4  23x2  36
51. P  x   3x 5  2x 4  9x3  6x2  12x  8
APPLICATIONS AND OTHER CHALLENGING PROBLEMS
1. A charitable organization wants to produce cookbooks to earn money to support their
charity. If x hundred cookbooks are sold, revenue and cost are given by
R  x   100x and C  x   x2  48x  100
Determine the profit functions (hint: P=R-C)
Determine the number of cooks that need to be sold to break even (hint: breakeven is where
P=0)
2. Using the profit equation P  x   x 3  5x2  3x  6 , when will the company break even if x
represents the units sold?
3. If the profit function of a given company can be modeled as a polynomial with all imaginary
zeros and the leading coefficient is positive. Would you invest in this company? Explain.
4. If the profit function of a given company can be modeled as a polynomial with all imaginary
zeros and the leading coefficient is negative. Would you invest in this company? Explain.
5. If the profit function of a given company can be modeled as a third degree polynomial with
only one real positive real zero at x=5 (x is the units sold) and the leading coefficient is
negative. Would you invest in this company? Explain.
6. If the profit function of a given company can be modeled as a third degree polynomial with
only one real positive real zero at x=5 (x is the units sold) and the leading coefficient is
positive. Would you invest in this company? Explain.
7. Is x+b a factor of x 3   2b  a  x 2   b2  2ab  x  ab2 ?
9. Divide x 3n  x2n  x n  1 by x n  1 .
8. Is x+b a factor of x 4   b2  a2  x 2  a2b2 ?
10. Divide x 3n  5x 2n  8 x n  4 by x n  1 .
11. Given that x=a is a zero of P  x   x 3   a  b  c  x2  ab  ac  bc  x  abc , find the other
two zeros given that a, b, c are real numbers and a>b>c.
12. Given that x=a is a zero of P  x   x 3   a  b  c  x2  ab  bc  ac  x  abc , find the other
two zeros given that a, b, c are real positive numbers.
13. Is it possible for an odd-degreed polynomial to have all imaginary complex zeros? Explain
14. Is it possible for an even-degree polynomial to have all imaginary zeros? Explain
15. Find a polynomial function that has degree 6 for which bi is a zero of multiplicity 3.
16. Find a polynomial function that has degree 4, for which a+bi is a zero of multiplicity 2.
TRUE/FALSE
17. A fifth-degree polynomial divided by a third-degree polynomial will yield a quadratic
quotient.
18. A third-degree polynomial divided by a linear polynomial will yield a linear quotient.
19. Synthetic division can be used whenever the degree of the dividend is exactly one more
than the degree of the divisor.
20. When the remainder is zero, the divisor is a factor of the dividend.
21. All real zeros of a polynomial correspond to x-intercepts.
22. A polynomial of degree n, n>0 must have at least one zero.
23. A polynomial of degree n, n>0, can be written as a product of n linear factors.
24. If x=1 is a zero of a polynomial function, than x=-1 is also a zero of the function.
CATCH THE MISTAKE
25. Determine whether x  2 is a factor of P  x   x 3  2x2  5x  6 .
2
1 2 5 6
SOLUTION
2 8 6 ,
1 4 3 0
This is incorrect, what mistake was made?
yes x  2 is a factor of P(x).
26. Given that 1 is a zero of P  x   x 3  2x2  7x  6 find all other zeros
SOLUTIONS:
Step 1: P(x) is a third degree polynomial, so we expect three zeros.
Step 2: Because 1 is a zero, -1 is a zero, so two linear factors are (x-1) and (x+1).
Step 3: Write the polynomial as a product of three linear factors
P(x)=(x-1)(x+1)(x-c)=(x2-1)(x-c)
Step 4: To find the remaining linear factor divide P(x) by x2-1. Which has a nonzero remainder. What went wrong?