College Algebra, 2013a

266 Polynomial Functions
In Exercises 31 - 40, you are given a polynomial and one of its zeros. Use the techniques in this
section to find the rest of the real zeros and factor the polynomial.
31. x 3 − 6x 2 +11x − 6, c = 1 32. x 3 − 24x 2 + 192x − 512, c =8
33. 3x 3 +4x 2 − x − 2, c = 2 3
34. 2x 3 − 3x 2 − 11x +6, c = 1 2
35. x 3 +2x 2 − 3x − 6, c = −2 36. 2x 3 − x 2 − 10x +5, c = 1 2
37. 4x 4 − 28x 3 +61x 2 − 42x +9,c = 1 2
is a zero of multiplicity 2
38. x 5 +2x 4 − 12x 3 − 38x 2 − 37x − 12, c = −1 is a zero of multiplicity 3
39. 125x 5 − 275x 4 − 2265x 3 − 3213x 2 − 1728x − 324, c = − 3 5
is a zero of multiplicity 3
40. x 2 − 2x − 2, c =1− √ 3
In Exercises 41 - 45, create a polynomial p which has the desired characteristics. You may leave
the polynomial in factored form.
41. ˆ The zeros of p are c = ±2 andc = ±1
ˆ The leading term of p(x) is 117x 4 .
42. ˆ The zeros of p are c = 1 and c =3
ˆ c = 3 is a zero of multiplicity 2.
ˆ The leading term of p(x) is−5x 3
43. ˆ The solutions to p(x) =0arex = ±3 andx =6
ˆ The leading term of p(x) is7x 4
ˆ The point (−3, 0) is a local minimum on the graph of y = p(x).
44. ˆ The solutions to p(x) =0arex = ±3, x = −2, and x =4.
ˆ The leading term of p(x) is−x 5 .
ˆ The point (−2, 0) is a local maximum on the graph of y = p(x).
45. ˆ p is degree 4.
ˆ as x →∞, p(x) →−∞
ˆ p has exactly three x-intercepts: (−6, 0), (1, 0) and (117, 0)
ˆ The graph of y = p(x) crosses through the x-axis at (1, 0).
46. Find a quadratic polynomial with integer coefficients which has x = 3 5 ± √
29
5
as its real zeros.