Q: Which of the following is the correct end behavior for $$ f(x) = -x^4$ + 7x^3$ + 4x^2$ + 32 $$?

$$ \lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = -\infty $$

Question 1

What is the simplified form of $$ (7 - i)(2 + 3i) $$?

Accepted Answer

$$ 14 + 21i - 2i - 3i^2$ = 14 + 19i + 3 = 17 + 19i $$

Question 2

Simplify $$ (5 + 4i)(5 - 4i) .$$

Accepted Answer

$$ 25 - 16i^2$ = 25 + 16 = 41 $$

Question 3

What is the value of $$ (-2 + \sqrt{-100})^2$ $$?

Accepted Answer

$$ (-2 + 10i)^2$ = (-2)^2$ + 2(-2)(10i) + (10i)^2$ = 4 - 40i - 100 = -96 - 40i $$

Question 4

Simplify $$ \frac{3 + 4i}{4 - 2i} $$ and write your answer in the form $$ a + bi .$$

Accepted Answer

$$ \frac{3 + 4i}{4 - 2i} \cdot \frac{4 + 2i}{4 + 2i} = \frac{(3 + 4i)(4 + 2i)}{(4 - 2i)(4 + 2i)} = \frac{12 + 6i + 16i + 8i^2$}{16 + 8i - 8i - 4i^2$} = \frac{12 + 22i - 8}{16 + 4} = \frac{4 + 22i}{20} = \frac{1}{5} + \frac{11}{10}i $$

Question 5

Simplify $$ \sqrt{-32} - \sqrt{-18} .$$

Accepted Answer

$$ \sqrt{-32} - \sqrt{-18} = 4i\sqrt{2} - 3i\sqrt{2} = i\sqrt{2} $$

Question 6

Which of the following is the correct end behavior for $$ f(x) = -x^4$ + 7x^3$ + 4x^2$ + 32 $$?

Accepted Answer

$$ \lim_{x 	o \infty} f(x) = -\infty, \quad \lim_{x 	o -\infty} f(x) = -\infty $$

Question 7

Divide $$ (4x^3$ + 6x^2$ + 3x - 1)  by$$ $$ (2x^2$ + 1) . $$What is the quotient?

Accepted Answer

$$ 2x + 3 + \frac{3x - 4}{2x^2$ + 1} $$

Question 8

Given $$ P(x) = 4x^3$ + 5x^2$ - 16x - 20 $$ and $$ x = -2  is a $$zero, which of the following is a factor of $$ P(x) $$?

Accepted Answer

$$ x + 2 $$

Question 9

Write a cubic polynomial with zeros at $$ x = -2 $$ and $$ x = 3 . $$Which of the following is the correct expanded form?

Accepted Answer

$$ (x + 2)(x - 3)(x - a) $$ for any real $$ a $$

Question 10

Solve the inequality $$ 2x^3$ - 7x \geq 12 - 7x^2$ . $$Which of the following intervals is part of the solution?

Accepted Answer

$$ x \leq -2 $$

Question 11

If $$ f(x) = 2x^2$ + x - 15 $$, for which values of $$ x  is$$ $$ f(x) \u003e 0 $$?

Accepted Answer

$$ x  2.5 $$

Question 12

Which of the following is the correct factorization of $$ f(x) = x^4$ + 2x^3$ + 6x^2$ + 18x - 27 $$ into linear factors?

Accepted Answer

$$ (x + 3)(x - 1)(x^2$ + 2x + 9) $$

Question 13

Which of the following is a possible set of zeros for a fourth degree polynomial with real coefficients if $$ x = 1 - 2i $$ and $$ x = i $$ are zeros?

Accepted Answer

$$ x = 1 - 2i, x = 1 + 2i, x = i, x = -i $$

Question 14

Which of the following describes the end behavior of $$ f(x) = 4x^5$ + 4x^3$ - x^6$ $$?

Accepted Answer

$$ \lim_{x 	o \infty} f(x) = -\infty, \quad \lim_{x 	o -\infty} f(x) = \infty $$

Question 15

Which of the following is the correct solution to $$ 4x^2$ \leq 1 - 2x $$?

Accepted Answer

$$ -\frac{1}{2} \leq x \leq \frac{1}{2} $$

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