Question 1

Solve the inequality $$36(x^2 - 1$$) \u003e 65x$$. Which of the following represents the solution set?

Accepted Answer

$$(-\infty, -\frac{9}{4}) \cup (\frac{9}{4}, \infty)$$

Question 2

Find all zeros of $$f(x) = x^3$ + 12x^2$ + 46x + 52$$ and write the polynomial as a product of linear factors.

Accepted Answer

$$f(x) = (x + 2)(x + 5 - i)(x + 5 + i)$$

Question 3

Given $$f(x) = x^4$ + kx^3$ + 2$$, and knowing that $$x + 1 is a $$factor, what is the value of $$k$$?

Accepted Answer

$$k = 3$$

Question 4

Form a cubic polynomial with zeros $$-3$$, $$-4$$, and $$4$$ and a leading coefficient of $$1. $$Which is correct?

Accepted Answer

$$f(x) = (x + 3)(x + 4)(x - 4)$$

Question 5

Form a degree 3 polynomial with real coefficients and zeros $$-2$$ and $$3 + i. $$Which is correct?

Accepted Answer

$$f(x) = (x + 2)[(x - 3)^2$ + 1]$$

Question 6

Use the Rational Zeros Theorem to list all possible rational zeros for $$f(x) = 2x^3$ - x^2$ - 14x + 7.$$

Accepted Answer

$$\pm1, \pm7, \pm\frac{1}{2}, \pm\frac{7}{2}$$

Question 7

Determine the real zeros and their multiplicities for $$f(x) = 4(x - 1)(x + 3)^2$. Does the graph touch or cross the x-axis at each zero?

Accepted Answer

Zero at $$x = 1 ($$multiplicity 1, crosses); zero at $$x = -3 ($$multiplicity 2, touches)

Question 8

List the potential rational zeros of $$f(x) = -4x^4$ + 3x^2$ - 4x + 6$$ using the Rational Zeros Theorem.

Accepted Answer

$$\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{6}{4}$$

Question 9

Find the y-intercept and x-intercepts of $$f(x) = 2x^3$ - x^2$ - 14x + 7.$$

Accepted Answer

y-intercept: $$(0, 7)$$; x-intercepts: $$(\frac{1}{2}, 0), (\sqrt{7}, 0), (-\sqrt{7}, 0)$$

Question 10

Find the real solutions to $$3x^3$$ - $$19x^2 + 30x - 8 = 0$.

Accepted Answer

$$x = \frac{1}{3}, 2, 4$$

Question 11

Find the y-intercept and x-intercepts of $$f(x) = -x^2$(x + 3)(x^2$ - 1).$$

Accepted Answer

y-intercept: $$(0, 0)$$; x-intercepts: $$(0, 0), (-3, 0), (1, 0), (-1, 0)$$

Question 12

Given $$f(x) = x^3$ - 2x^2$ - 11x + 52$$ and knowing $$-4 is a $$zero, find the remaining zeros.

Accepted Answer

$$3 + 2i, 3 - 2i$$

Question 13

Determine the real zeros and their multiplicities for $$f(x) = 4(x^2$ + 3)(x + 2)^2$. Does the graph touch or cross the x-axis at each zero?

Accepted Answer

Zero at $$x = -2 ($$multiplicity 2, touches)

Question 14

Given $$f(x) = 2x^4$ - 15x^3$ + 60x^2$ - 115x + 78$$ and knowing $$2 + 3i is a $$zero, what are the remaining zeros?

Accepted Answer

$$2 - 3i, \frac{3}{2}, 2$$

Question 15

Find all zeros of $$f(x) = x^3$ - 2x^2$ - 5x + 6$$ and write the polynomial as a product of linear factors.

Accepted Answer

$$f(x) = (x + 3)(x + 2)(x - 3)$$

Question 16

Solve the inequality $$x^2 - 36$$ \u003e 0$$. Which interval represents the solution?

Accepted Answer

$$(-\infty, -6) \cup (6, \infty)$$

Question 17

A chemical reaction is modeled by $$f(x) = x^4$ + 6x^3$ + 12x^2$ + 24x + 32. $$Find all zeros and write the polynomial as a product of linear factors.

Accepted Answer

$$f(x) = (x + 2)^4$

Precalculus Polynomial and Inequality Problem-Solving Guidance

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