Q: Divide $$x^2 + 7x + 2$ by x + 2$. What is the quotient?

$$x + 5$$ with a remainder of $$-8$$,\text{ so }\frac{$$x^2 + 7x + 2$$}{x + 2} = x + 5 + \frac{-8}{x + 2}$

Question 1

Which of the following best describes the first step in dividing a polynomial by a binomial using long division?

Accepted Answer

Arrange both the dividend and divisor in descending powers of $$x.$$

Question 2

Divide $$x^2 + 7x + 2$ by x + 2$. What is the quotient?

Accepted Answer

$$x + 5$$ with a remainder of $$-8$$,	ext{ so }\frac{$$x^2 + 7x + 2$$}{x + 2} = x + 5 + \frac{-8}{x + 2}$

Question 3

When dividing $$2x^3$$ + $$3x^2 - x + 4$ by x - 1$, what is the first term of the quotient?

Accepted Answer

$$2x^2$$

Question 4

A chemist is mixing solutions and models the total concentration as $$4x^2 + 8x + 12$. If each container holds x + 2$ units, how many full containers can be filled, and what is the leftover concentration?

Accepted Answer

Quotient: $$4x + 0$$, Remainder: $$12$$; so $$4x^2 + 8x + 12 = (x + 2)(4x) + 12$

Question 5

Divide $$x^3 - 4x + 5$ by x - 2$. What is the quotient and remainder?

Accepted Answer

Quotient: $$x^2 + 2x + 0$, Remainder: 9$

Question 6

Which of the following is the correct order of steps in polynomial long division?

Accepted Answer

Divide, multiply, subtract, bring down

Question 7

If you are dividing $$3x^2 + 0x - 9$ by x - 3$, what is the importance of writing 0x$ in the dividend?

Accepted Answer

It ensures all like terms are aligned during the division.

Question 8

Divide $$2x^2 + 5x + 3$ by x + 1$. What is the quotient?

Accepted Answer

$$2x + 3$$ with a remainder of $$0$$, so $$2x^2 + 5x + 3 = (x + 1)(2x + 3$$)$$

Question 9

A company models its profit as $$x^3$$ + $$2x^2 - x + 6$. If each product batch is represented by x + 3$, what is the quotient and remainder when dividing the profit by the batch size?

Accepted Answer

Quotient: $$x^2 - x + 2$, Remainder: 0$, so x^3$ + 2x^2$ - x + 6 = (x + 3)(x^2$ - x + 2)$$

Question 10

Divide $$x^3$$ + $$0x^2 - 4x + 8$ by x - 2$. What is the quotient?

Accepted Answer

$$x^2 + 2x + 4$

Question 11

Which of the following statements is TRUE about the remainder in polynomial long division?

Accepted Answer

The remainder has a degree less than the divisor.

Question 12

Divide $$2x^3$$ + $$3x^2 - x + 4$ by x - 1$. What is the complete answer in the form $$	ext{Quotient} + \frac{	ext{Remainder}}{	ext{Divisor}}$$?

Accepted Answer

$$2x^2 + 5x + 4$$ + \frac{8}{x - 1}$$

Question 13

A medical researcher models the spread of a virus as $$x^2 + 6x + 8$. If each group of patients is represented by x + 2$, how many groups can be formed and what is the leftover?

Accepted Answer

Quotient: $$x + 2$$, Remainder: $$4$$

Question 14

When dividing $$x^2 - 5x + 6$ by x - 2$, what is the quotient?

Accepted Answer

$$x - 3$$

Question 15

Divide $$3x^2 + 0x - 9$ by x - 3$. What is the quotient and remainder?

Accepted Answer

Quotient: $$3x + 9$$, Remainder: $$-27$$, so $$3x^2 + 0x - 9 = (x - 3)(3x + 9) - 27$

Question 16

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

Accepted Answer

Multiplying the divisor by the entire dividend

Question 17

Divide $$x^3$$ + $$2x^2 - x + 6$ by x + 3$. What is the remainder?

Accepted Answer

$$6$$

Question 18

A factory produces widgets modeled by $$2x^2 + 5x + 3$. If each box holds x + 1$ widgets, how many full boxes can be packed and how many widgets are left over?

Accepted Answer

Quotient: $$2x + 3$$, Remainder: $$0$$

Dividing Polynomials by Binomials: Long Division Method

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