Question 1

Which of the following is the correct first step when factoring the polynomial $$x^4$$ + $$2x^2 - 8$?

Accepted Answer

Let $$u = x^2$ and rewrite as u^2$ + 2u - 8.$$

Question 2

Given the equation $$x^4$$ - $$5x^2 + 4 = 0$, what are all real solutions?

Accepted Answer

$$x = 2, x = -2, x = 1, x = -1$$

Question 3

If a polynomial equation has a factor $$(x-3)^2$$, what does this indicate about the graph at $$x=3$$?

Accepted Answer

The graph touches but does not cross the x-axis at $$x=3.$$

Question 4

Solve for $$x$$: $$\sqrt{2x+3} = x-1$$

Accepted Answer

$$x = 3$$

Question 5

Which of the following is a valid step when solving $$\sqrt{x+5} + 2 = x$$?

Accepted Answer

Isolate the square root: $$\sqrt{x+5} = x-2.$$

Question 6

What is the solution to $$x^2 + 4 = 0$?

Accepted Answer

$$x = 2i, x = -2i$$

Question 7

Given $$x^4 - 81 = 0$, which of the following is the correct factorization?

Accepted Answer

$$(x-3)(x+3)(x^2+9$$)$$

Question 8

If $$x = 5 is a $$solution of multiplicity 2 for a polynomial, what does this mean?

Accepted Answer

The graph touches but does not cross the x-axis at $$x=5.$$

Question 9

Solve for $$x$$: $$\sqrt{3x+1} = x$$

Accepted Answer

$$x = \frac{1 + \sqrt{5}}{2}$$

Question 10

Which of the following best describes the process for solving equations involving square roots?

Accepted Answer

Isolate the radical, square both sides, solve, and check for extraneous solutions.

Question 11

What is the solution to $$x^2 - 6x + 13 = 0$?

Accepted Answer

$$x = 3 \pm 2i$$

Question 12

Which of the following is a possible extraneous solution when solving $$\sqrt{x+2} = x-2$$?

Accepted Answer

$$x = 2$$

Question 13

If $$x^2 = -9$, what are the solutions?

Accepted Answer

$$x = 3i, x = -3i$$

Question 14

Which of the following is the correct solution to $$x^4$$ - $$16x^2 = 0$?

Accepted Answer

$$x = 0, x = 4, x = -4$$

Question 15

What is the correct interpretation of a solution with multiplicity 1?

Accepted Answer

The graph crosses the x-axis at that solution.

Question 16

Solve for $$x$$: $$\sqrt{2x-1} = x$$

Accepted Answer

$$x = \frac{1 + \sqrt{5}}{2}$$

Question 17

Which of the following is the correct factorization for $$x^4$$ + $$2x^2 - 8$?

Accepted Answer

$$(x^2$$ + $$4)(x^2 - 2$$)$$

Question 18

If $$x^2 + 6x + 13 = 0$, what are the solutions?

Accepted Answer

$$x = -3 \pm 2i$$

Question 19

Which of the following is a necessary step when checking for extraneous solutions in radical equations?

Accepted Answer

Substitute each solution back into the original equation.

Solving Miscellaneous Polynomial and Radical Equations

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