Question 1

According to the Rational Zero Theorem, what are the possible rational zeros of $$f(x) = 4x^5$ + 12x^4$ - x - 3$$?

Accepted Answer

$$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}$$

Question 2

Given $$f(x) = x^3$ + 2x^2$ - 5x - 6$$, which of the following is NOT a possible rational zero according to the Rational Zero Theorem?

Accepted Answer

$$4$$

Question 3

If $$f(x) = x^3$ + 7x^2$ + 11x - 3$$ and you use synthetic division with $$x = -3$$, what is the remainder?

Accepted Answer

$$0$$

Question 4

What is the factorization of $$f(x) = x^3$ + 7x^2$ + 11x - 3$$ after finding $$x = -3 is a $$zero?

Accepted Answer

$$f(x) = (x + 3)(x^2$ + 4x - 1)$$

Question 5

Using the quadratic formula, what are the remaining zeros of $$x^2 + 4x - 1 = 0$?

Accepted Answer

$$x = -2 \pm \sqrt{5}$$

Question 6

Given $$f(x) = x^4$ - 6x^2$ - 8x + 24$$, what is the list of possible rational zeros?

Accepted Answer

$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$$

Question 7

If $$x = 2 is a $$zero of $$f(x) = x^4$ - 6x^2$ - 8x + 24$$, what is the resulting depressed cubic after synthetic division?

Accepted Answer

$$x^3$$ + $$2x^2 - 2x - 12$

Question 8

What are the complex roots of $$x^2 + 4x + 6 = 0$?

Accepted Answer

$$x = -2 \pm i\sqrt{2}$$

Question 9

How many positive real zeros does $$f(x) = x^3$ + 2x^2$ + 5x + 4$$ have, according to Descartes’ Rule of Signs?

Accepted Answer

0

Question 10

Given $$f(x) = x^3$ + 2x^2$ + 5x + 4$$, what is $$f(-x)$$?

Accepted Answer

$$-x^3$$ + $$2x^2 - 5x + 4$

Question 11

If $$f(x) = x^3$ + 2x^2$ + 5x + 4$$ and $$x = -1 is a $$zero, what is the quadratic factor that remains?

Accepted Answer

$$x^2 + x + 4$

Question 12

A polynomial $$f(x) of $$degree 4 has zeros at $$-2, 2, i. If$$ $$f(3) = -150$$, what is $$f(x)$$?

Accepted Answer

$$f(x) = -3x^4$ + 9x^2$ + 12$$

Question 13

If $$f(x) = 3x^7$ - 2x^5$ - x^4$ + 7x^2$ + x - 3$$, how many sign changes are there in $$f(x)$$ and what does this tell you about the number of positive real zeros?

Accepted Answer

3 sign changes; 3 or 1 positive real zeros

Question 14

Given $$f(x) = 3x^7$ - 2x^5$ - x^4$ + 7x^2$ + x - 3$$, how many sign changes are there in $$f(-x)$$ and what does this tell you about the number of negative real zeros?

Accepted Answer

4 sign changes; 4, 2, or 0 negative real zeros

Question 15

If a polynomial has a root $$a + bi$$, what must also be true about its roots if the coefficients are real?

Accepted Answer

The conjugate $$a - bi is $$also a root

Question 16

What is the Linear Factorization Theorem for a degree $$n$$ polynomial $$f(x)$$ with leading coefficient $$a_n$$?

Accepted Answer

$$f(x) = a_n(x - c_1$)(x - c_2$)\dots(x - c_n)$$ where $$c_1$$, \dots, c_n$$ are the complex roots

Question 17

If $$f(x) = x^4$ - 6x^2$ - 8x + 24$$ and $$x = 2 is a $$zero of multiplicity 2, what does this mean about the graph at $$x = 2$$?

Accepted Answer

The graph touches and turns around at $$x = 2$$

Rational Zero Theorem and Roots of Polynomial Functions

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