Question 1

Which of the following is the correct first step when performing long division of polynomials?

Accepted Answer

Arrange the terms of both the dividend and the divisor in descending powers of any variable.

Question 2

During long division of polynomials, after dividing the first term of the dividend by the first term of the divisor, what is the next step?

Accepted Answer

Multiply every term in the divisor by the first term in the quotient.

Question 3

In the process of long division of polynomials, when do you stop repeating the steps of division, multiplication, and subtraction?

Accepted Answer

When the degree of the remainder is less than the degree of the divisor.

Question 4

Given the division $$\$$ \frac{$$2x^3$$ + $$3x^2 - x + 5$$}{x - 2} \$ using long division, what is the first term of the quotient?

Accepted Answer

$$\$$ $$2x^2$$ \$

Question 5

If you are dividing $$\$$ $$x^3$$ + $$2x^2 - 5x + 6$$ \$$ by $$\$$ x - 1 \$$ using long division, what do you do after multiplying $$\ x - 1$$ \$ by the first term of the quotient?

Accepted Answer

Subtract the product from the dividend.

Question 6

In long division of polynomials, what is the purpose of 'bringing down' the next term?

Accepted Answer

To form a new dividend for the next step of division.

Question 7

When dividing $$\$$ $$3x^4$$ - $$2x^3$$ + $$x^2 - 5$$ \$$ by $$\$$ x^2$ + 1 \$$ using long division, what is the degree of the first term in the quotient?

Accepted Answer

$$\ 2$$ \$

Question 8

Which of the following best describes the final remainder in polynomial long division?

Accepted Answer

Its degree is less than the degree of the divisor.

Question 9

If the remainder after dividing $$\$$ $$4x^3$$ + $$2x^2 - x + 7$$ \$$ by $$\$$ x + 1 \ is$$ $$\ 3$$ \$, how can you express the result of the division?

Accepted Answer

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

Question 10

A real-world application: A company’s profit, $$\ P(x$$) = $$2x^3$$ - $$5x^2 + 4x - 8$$ \$$, is to be divided by $$\$$ x - 2 \ to $$analyze profit per unit. What is the first step in using long division to find the profit per unit?

Accepted Answer

Divide $$\$$ $$2x^3$$ \$$ by $$\$$ x \ to $$get $$\$$ $$2x^2$$ \$.

Question 11

Which of the following is NOT a step in the long division of polynomials?

Accepted Answer

Add the product to the dividend.

Question 12

If you divide $$\$$ $$x^3$$ + $$4x^2 + 5x + 2$$ \$$ by $$\$$ x + 2 \$$ and the remainder is zero, what does this indicate?

Accepted Answer

$$\ x + 2$$ \$ is a factor of the dividend.

Question 13

In a chemical process, the concentration of a substance is modeled by $$\ C(x$$) = $$6x^3$$ - $$9x^2 + 2x - 4$$ \$. If you want to find the average rate of change per unit, which division would you perform?

Accepted Answer

Divide $$\ C(x$$) \$$ by $$\$$ x \$$ using long division.

Question 14

When using long division, what should you do if a term is missing in the dividend (e.g., no $$\$$ $$x^2$$ \$ term)?

Accepted Answer

Insert a zero coefficient for the missing term.

Division of Polynomials: Long Division and Synthetic Division

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