Quiz worksheet on modified long division with six problems involving polynomial expressions.

Let’s solve each problem one by one, step by step.

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**Problem 1: Which term is missing in this problem?
(2x³ + 5x² + 9) ÷ (x + 3)**

When doing polynomial long division, we need to make sure all powers of x are present — even if their coefficient is zero. The dividend is:

→ 2x³ + 5x² + 0x + 9

We’re missing the x term (which would be “+ 0x”). So the missing term is just x.

✔ Answer: B. x

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Problem 2: Divide: (2x³ - 5x² + 3x + 7) ÷ (x - 2)

We’ll use synthetic division since divisor is linear (x - 2).

Set up synthetic division with root = 2:

Coefficients: 2 | -5 | 3 | 7

Bring down 2.

Multiply 2 × 2 = 4 → add to next term: -5 + 4 = -1
Multiply -1 × 2 = -2 → add to next term: 3 + (-2) = 1
Multiply 1 × 2 = 2 → add to last term: 7 + 2 = 9

So quotient is: 2x² - x + 1, remainder 9.

That means answer is: 2x² - x + 1 + 9/(x - 2)

✔ Answer: C. 2x² - x + 1 + (9/x-2)

*(Note: It should technically be written as 9/(x - 2), but option C matches that intent.)*

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Problem 3: Divide: (4x² - 24x + 35) ÷ (2x - 5)

Try factoring numerator or do long division.

Let’s try factoring:

Look for two numbers that multiply to 4×35=140 and add to -24 → -10 and -14.

Split middle term:

4x² -10x -14x +35
= 2x(2x - 5) -7(2x - 5)
= (2x - 7)(2x - 5)

So dividing by (2x - 5) gives: 2x - 7

✔ Answer: A. 2x - 7

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Problem 4: Divide: (4x³ - 2x² - 3) by (2x² - 1)

This is polynomial long division.

Divide leading terms: 4x³ ÷ 2x² = 2x

Multiply (2x)(2x² - 1) = 4x³ - 2x

Subtract from original:

(4x³ - 2x² - 3) - (4x³ - 2x) = -2x² + 2x - 3

Now divide -2x² ÷ 2x² = -1

Multiply (-1)(2x² - 1) = -2x² + 1

Subtract:

(-2x² + 2x - 3) - (-2x² + 1) = 2x - 4

So quotient is 2x - 1, remainder 2x - 4.

Answer: 2x - 1 + (2x - 4)/(2x² - 1)

✔ Answer: C. 2x - 1 + (2x - 4)/(2x² - 1)

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Problem 5: Divide: (x³ + 3x² - 4x - 12) by (x² + x - 6)

Do polynomial long division.

First term: x³ ÷ x² = x

Multiply x(x² + x - 6) = x³ + x² - 6x

Subtract:

(x³ + 3x² - 4x - 12) - (x³ + x² - 6x) = (2x² + 2x - 12)

Next term: 2x² ÷ x² = 2

Multiply 2(x² + x - 6) = 2x² + 2x - 12

Subtract:

(2x² + 2x - 12) - (2x² + 2x - 12) = 0

Perfect! No remainder.

Quotient is: x + 2

✔ Answer: B. x + 2

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Final Answer:
1. B
2. C
3. A
4. C
5. B