Math 121, Practice Questions for Chapter 3
1. Use long division to find (6x4+ 8x3−47x2+ 19x+ 5) ÷(2x2+ 6x−5).
2. Use synthetic division to find (x4+ 9x3−5x+ 10) ÷(x+ 3)
3. Let P(x) = 6x4+ 8x3−8x2−31x+ 4, use the remainder theorem to find P(6).
4. Determine the far right and far left behavior of the polynomials:
(a) P(x) = −3x6+ 20x5+ 7x2−12.
(b) P(x) = x3+ 5x+ 7.
(c) P(x) = 22x8+ 2000000x6−4x9.
(d) P(x) = 22x8+ 2000000x6−4x7.
5. (a) Find the x-intercepts of P(x)=(x−3)2(x−2)4(x+ 1)3(x+ 3). For each intercept,
determine whether the graph of P(x) crosses or merely touches the x-axis.
(b) What is the degree of P(x)? Determine the far right and far left behavior of P(x).
(c) Using the information above, sketch a very crude graph of P(x).
(d) Find the x-intercepts of P(x)=(x−3000)3(x−2000)2(x+ 1000)1(x+ 3000)2000 . For each
intercept, determine whether the graph of P(x) crosses or merely touches the x-axis.
6. Consider the polynomial P(x) = 2x5−3x4+ 12x2+ 13x−6.
(a) Use the Rational Zero Theorem to list the possible rational zeros of P(x)
(b) State the Intermediate Value Theorem to prove there is a zero between x= 0 and x= 1.
Do not find that zero.
(c) Using Decartes’ Rule of Signs, determine the number of possible positive and negative real
zeros of P(x).
7. (a) Using the information that the polynomial P(x)=2x4+ 7x3+ 5x2+ 7x+ 3 has zeros
−1
2and −3, find all zeros of P(x) and write P(x) as a product of linear factors.
(b) Given that 2 + 3iis a zero of
Q(x) = x4−4x3+ 14x2−4x+ 13
find the remaining zeros, and write Q(x) as a product of linear factors.
8. (a) Find a polynomial with real coefficients of smallest degree that has zeros 2 +3i,−2 and
3.
(b) With help from (a), find a polynomial P(x) with real coefficients of smallest degree that
has zeros 2 + 3i,−2 and 3 such that P(1) = 120.
9. Find the x-intercepts, y-intercepts, horizontal, vertical and slant asymptotes (if they exist)
and determine the left and right behavior near the vertical asymptotes for:
1
