How many x intercepts appear on the graph of this polynomial function?
f(x) = x^4 - x^3 + x^2 - x
a) 1 x intercept
b) 2 x intercepts
c) 3 x intercepts
d) 4 x intercepts

The correct answer is B.

When you factor the polynomial you get the following:

f(x) = (x^2 + 1)(x)(x - 1)

The last two parenthesis give you the intercepts of 0 and 1. The first parenthesis will give you two non-real zeros, but these are not visible in a graph as the question asks.

From the attachment, the solution set of each combined ineaquality are as follows:

Part 1:

For 1,
The solution set for x > -4  and x ≤ -1 is the number line labeled I.

Part 2:

For 2,
x + 5 > 4 and x - 2 < 2
x + 5 - 5 > 4 - 5 and x - 2 + 2 < 2 + 2
x > -1 and x < 4
The solution set is the number line labelled O

Part 3:

For 3,
The solution set for y ≤ -2 or y > 3 is the number line labelled O.

Part 4:

For 4,
-3t > 12 or 5t ≥ 10
-3t / -3 < 12 / -3 or 5t /5 ≥ 10 / 5         [note that when you divide an ineduality term by a negative number the inequality sign changes]
t < -4 or t ≥ 2
The solution set is the number line labelled E.

Part 5:
2n + 5 > 1 and 3n + 4 > 7
2n + 5 - 5 > 1 - 5 and 3n + 4 - 4 > 7 - 4
2n > -4 and 3n > 3
2n / 2 > -4 / 2 and 3n / 3 > 3 / 3
n > -2 and n > 1

The solution set is the number line labelled N.

Part 6:

For 6,
-4u + 9 > 1 and 7u - 13 ≤ -6
-4u + 9 - 9 > 1 - 9 and 7u - 13 + 13 ≤ -6 + 13
-4u > -8 and 7u ≤ 7
-4u / -4 < -8 / -4 and 7u / 7 ≤ 7 / 7
u < 2 and u ≤ 1

The solution set is the number line labelled G.

Part 7:

For 7,
32 ≤ 3x + 20 or 17 > 1 - 8x
32 - 20 ≤ 3x + 20 - 20 or 17 - 1 > 1 - 8x - 1
12 ≤ 3x or 16 > -8x
12 / 3 ≤ 3x / 3 or 16 / -8 < -8x / -8
4 ≤ x or -2 < x

The solution set is the number line labelled M.

Part 8:

For 8,
-2k + 8 < 14 or 3k + 1 < 1
-2k + 8 - 8 < 14 - 8 or 3k + 1 - 1 < 1 - 1
-2k < 6 or 3k < 0
-2k / -2 > 6 / -2 or 3k / 3 < 0 / 3
k > -3 or k < 0.

The solution set is the number line labelled H.

Part 9:

For 9,
5(w + 4) ≥ 5  and 2(w + 4) < 12
5(w + 4) / 5 ≥ 5 / 5 and 2(w + 4) < 12 / 2
w + 4 ≥ 1 and w + 4 < 6
w + 4 - 4 ≥ 1 - 4 and w + 4 - 4 < 6 - 4
w ≥ -3 and w < 2

The solution set is the number line labelled T.

Part 10:

For 10,
3(6 - y) ≤ 6 and 6 - y ≥ 8
3(6 - y) / 3 ≤ 6 / 3 and 6 - y ≥ 8
6 - y ≤ 2 and 6 - y ≥ 8
6 - y - 6 ≤ 2 - 6 and 6 - y - 6 ≥ 8 - 6
-y ≤ -4 and -y ≥ 2
-y / -1 ≥ -4 / -1 and -y / -1 ≤ 2 / -1
y ≥ 4 and y ≤ -2

Because there is no number that is greater that 4 and at the same time less than -2. The solution set is empty.
Thus the solution set is labelled S.

Part 11:

For 11,
3x < 2x - 3 or 7x > 4x - 9
3x - 2x < 2x - 3 - 2x or 7x - 4x > 4x - 9 - 4x
x < -3 or 3x > -9
x < -3 or 3x / 3 > -9 / 3
x < -3 or x > -3

The solution set is the number line labelled P.

Part 12:

For 12,
x / 2 ≤ -2 or -x / 2 ≥ 0
2(x / 2) ≤ 2(-2) or 2(-x / 2) ≥ 2(0)
x ≤ -4 or -x ≥ 0
x ≤ -4 or -x / -1 ≤ 0 / -1
x ≤ -4 or x ≤ 0

The solution set is the number line labelled A.