Math 140. Test 2 (v1) (Fall 2005. Harvey).
Name (2 points):
No notes or texts allowed. You may use a TI-83, TI-84, TI-86 or equivalent calculator. Show all
work.
1. (8 points) Consider the parabola defined by the equation:
y= 3x2โ6x+ 1.
Find the coordinates of the vertex. Write the parabola in standard form: y=a(xโh)2+k.
2. (9 points) Let
p(x) = (x+ 4)3(2xโ3)4(5x+ 2)2
(1) What is the degree of p(x)? (2) What is the leading coefficient of p(x)? (3) Describe the limiting
behavior of p(x).
3. (6 points) What is the remainder of the division:
x4+ 4x3โ2x2+ 5x
xโ1
4. (9 points) Let
p(x) = (x+ 2)(2x+ 3)2(x+ 1)4
(1) List the zeros of p(x). (2) Give the multiplicity of each zero. (3) For which zeros will p(x) cross
through the x-axis and for which zeros will p(x) touch the axis but not cross through?
5. (10 points) Let
p(x) = x3+ 7x2+ 9x+ 3
Make a list of all possible rational zeros of p(x) (according to the Rational Zeros Theorem). Find
all the real zeros of p(x).
6. (10 points) Let
p(x) = x4โx3โ8x2+ 6x+ 12
Make a list of all possible rational zeros of p(x) (according to the Rational Zeros Theorem). Find
all the real zeros of p(x).
7. (8 points) Let
p(x) = x7+x6โx5+ 3x4โx3โx2โx+ 1
According to Descartes Rule of Signs, how many positive real zeros may p(x) have? How many
negative real zeros may p(x) have?
8. (8 points) Red pine trees are common in the northern United States. In fact, they are the
state tree of Minnesota. The following ages and heights of red pines is gathered:
