Question 1

Which of the following best describes the Division Algorithm for polynomials?

Accepted Answer

If $$f(x)$$ and $$d(x)$$ are polynomials, then $$f(x) = d(x)q(x) + r(x)$$, where $$r(x) is $$the remainder and $$q(x) is $$the quotient, and the degree of $$r(x) is $$less than the degree of $$d(x).$$

Question 2

What is the result of dividing $$2x^3$$ - $$3x^2 + 4x - 5$ by x - 2$ using synthetic division?

Accepted Answer

$$2x^2 + x + 6$ with a remainder of 7$

Question 3

If $$f(x) = x^3$ + 2x^2$ - 5x + 6$$, what is the remainder when $$f(x) is $$divided by $$x - 1$$?

Accepted Answer

$$f(1) = 1 + 2 - 5 + 6 = 4$$

Question 4

Which of the following statements is true according to the Factor Theorem?

Accepted Answer

If $$f(c) = 0$$, then $$x - c is a $$factor of $$f(x).$$

Question 5

A chemist models the concentration of a substance with $$f(x) = 3x^3$ - 2x^2$ + x - 4. $$What is $$f(-2)$$, and what does it represent if $$x is $$time in hours?

Accepted Answer

$$f(-2) = -44$$, representing the concentration at $$t = -2$$ hours.

Question 6

Which step is NOT part of the long division process for polynomials?

Accepted Answer

Add the product of the divisor and quotient to the dividend.

Question 7

Given $$f(x) = x^3$ - 4x^2$ + 5x - 2$$, and knowing that $$x = 1 is a $$zero, which of the following is a factor of $$f(x)$$?

Accepted Answer

$$x - 1$$

Question 8

A company’s profit is modeled by $$P(x) = 4x^3$ - 12x^2$ + 9x - 5. If$$ $$x = 3 is a $$solution to $$P(x) = 0$$, what does this mean in practical terms?

Accepted Answer

The company breaks even when $$x = 3$$ units are sold.

Question 9

Which of the following is the correct setup for synthetic division to divide $$2x^3$$ + $$3x^2 - x + 4$ by x - 2$?

Accepted Answer

Use $$c = 2$$ and the coefficients $$[2, 3, -1, 4].$$

Question 10

If $$f(x) = x^4$ - 2x^3$ + x - 5$$, what is the degree of the remainder when $$f(x) is $$divided by $$x^2 + 1$?

Accepted Answer

The degree is less than 2.

Question 11

A medical researcher uses $$f(x) = x^3$ - 6x^2$ + 11x - 6 to $$model the spread of a virus. If $$f(1) = 0$$, what does this indicate?

Accepted Answer

The spread rate is zero at $$x = 1$$, so $$x - 1 is a $$factor.

Question 12

Which of the following is a necessary step when using synthetic division if a term is missing in the dividend?

Accepted Answer

Insert a zero for the missing term’s coefficient.

Question 13

If $$f(x) = 2x^3$ - 5x^2$ + 3x - 7$$ and $$f(2) = 0$$, which statement is true?

Accepted Answer

$$x - 2 is a $$factor of $$f(x).$$

Question 14

A polynomial $$f(x) is $$divided by $$x - 3$$ and the remainder is $$5. $$What is $$f(3)$$?

Accepted Answer

$$5$$

Question 15

Which of the following best explains why the degree of the remainder is less than the degree of the divisor in polynomial division?

Accepted Answer

Because division stops when the remainder’s degree is less than the divisor’s degree.

Dividing Polynomials: Long Division, Synthetic Division, Remainder and Factor Theorems

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