Worksheet for practicing polynomial division with eight problems involving single-variable algebraic expressions.

Let’s solve each problem one by one. We’re dividing polynomials by monomials — that means we divide each term in the polynomial by the single term (monomial) outside.

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Problem 1:

(6u³ + 3u⁶ + 9u²) ÷ 3u

Break it into parts:

- 6u³ ÷ 3u = (6÷3) × u^(3-1) = 2u²
- 3u⁶ ÷ 3u = (3÷3) × u^(6-1) = 1u⁵ = u⁵
- 9u² ÷ 3u = (9÷3) × u^(2-1) = 3u¹ = 3u

Now put them together:
→ u⁵ + 2u² + 3u

*(Note: Usually we write terms from highest power to lowest, so u⁵ first.)*

✔ Final for #1: u⁵ + 2u² + 3u

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Problem 2:

(2k⁵ - 8k⁷ - 6k²) ÷ 2k²

Divide each term:

- 2k⁵ ÷ 2k² = (2÷2) × k^(5-2) = 1k³ = k³
- -8k⁷ ÷ 2k² = (-8÷2) × k^(7-2) = -4k⁵
- -6k² ÷ 2k² = (-6÷2) × k^(2-2) = -3k⁰ = -3×1 = -3

Put together:
→ -4k⁵ + k³ - 3

*(Again, order by descending powers: -4k⁵ + k³ - 3)*

✔ Final for #2: -4k⁵ + k³ - 3

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Problem 3:

(5m⁴ + 10m⁶ - 15) ÷ 5

This is simpler — just divide every coefficient by 5.

- 5m⁴ ÷ 5 = m⁴
- 10m⁶ ÷ 5 = 2m⁶
- -15 ÷ 5 = -3

Order by descending powers:
→ 2m⁶ + m⁴ - 3

✔ Final for #3: 2m⁶ + m⁴ - 3

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Problem 4:

(-8n⁶ - n⁴) ÷ (-n⁴)

Divide each term:

- -8n⁶ ÷ (-n⁴) = (-8 ÷ -1) × n^(6-4) = 8n²
- -n⁴ ÷ (-n⁴) = (-1 ÷ -1) × n^(4-4) = 1 × n⁰ = 1

So:
→ 8n² + 1

✔ Final for #4: 8n² + 1

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Problem 5:

(-4w⁷ + 6w⁸ + w³ - 6w⁵) ÷ w³

Divide each term by w³:

- -4w⁷ ÷ w³ = -4w^(7-3) = -4w⁴
- 6w⁸ ÷ w³ = 6w^(8-3) = 6w⁵
- w³ ÷ w³ = 1w⁰ = 1
- -6w⁵ ÷ w³ = -6w^(5-3) = -6w²

Now arrange in descending order of exponents:
→ 6w⁵ - 4w⁴ - 6w² + 1

✔ Final for #5: 6w⁵ - 4w⁴ - 6w² + 1

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Problem 6:

(-7x⁵ + 7x³) ÷ 7x⁴

Divide each term:

- -7x⁵ ÷ 7x⁴ = (-7÷7) × x^(5-4) = -1x¹ = -x
- 7x³ ÷ 7x⁴ = (7÷7) × x^(3-4) = 1x^(-1) = 1/x → but since this is a polynomial division and we usually expect polynomial answers, let’s check if we did right.

Wait — actually, x³ ÷ x⁴ = x^(-1), which is not a polynomial term. But maybe the problem allows rational expressions? Let’s see what the original says — “Divide the following.” It doesn’t specify only polynomial results.

But looking at other problems, they all result in polynomials. Did I misread?

Original: (-7x⁵ + 7x³) ÷ 7x⁴

Let me recompute:

First term: -7x⁵ / 7x⁴ = -x ✔

Second term: 7x³ / 7x⁴ = x³/x⁴ = 1/x ✘ Not a polynomial.

Hmm… perhaps there’s a typo? Or maybe we leave it as is?

Wait — let’s factor numerator first:

Numerator: -7x⁵ + 7x³ = 7x³(-x² + 1)

Denominator: 7x⁴

So: [7x³(1 - x²)] / (7x⁴) = (1 - x²)/x = 1/x - x

Still not a polynomial.

But looking back at the worksheet title: “Single Variable: S1” — likely expects simplified expression even if not polynomial.

Alternatively, maybe I made a mistake in reading the problem?

Wait — problem 6 is written as:
(-7x⁵ + 7x³) ÷ 7x⁴

Yes.

So answer is: -x + 1/x

But that’s unusual for this level. Maybe it's acceptable?

Alternatively, perhaps the divisor is 7x³? But no, it says 7x⁴.

Let me double-check with another approach.

Write as fractions:

= (-7x⁵)/(7x⁴) + (7x³)/(7x⁴)
= -x + 1/x

So mathematically correct.

But since other problems give polynomials, maybe this one is meant to be left as is? Or perhaps I should write it as a single fraction?

Actually, let’s keep it as two terms: -x + \frac{1}{x}

But for consistency with others, maybe the problem has a typo? However, as given, we must solve it correctly.

Wait — let me check problem 8: (6v⁶ - 5v⁴ + v²) ÷ v — that will also have negative exponents if divided directly? No:

v² ÷ v = v, so okay.

Back to #6: Perhaps the intended answer is -x + x^{-1}, but that’s not standard.

Alternatively, maybe the problem is designed to show that sometimes you get non-polynomial results? Unlikely for this level.

Wait — let me recalculate exponent subtraction:

x³ ÷ x⁴ = x^{3-4} = x^{-1} — yes.

Perhaps the problem meant 7x³ as divisor? Then:

If divisor were 7x³:

-7x⁵ ÷ 7x³ = -x²
7x³ ÷ 7x³ = 1
→ -x² + 1 — nice polynomial.

But the image clearly says 7x⁴.

Given that, I’ll go with the mathematical truth.

✔ Final for #6: -x + \frac{1}{x}

But wait — let me check online or think again. In many algebra classes, when dividing polynomials by monomials, if the degree of a term in the dividend is less than the divisor, you still write it as a fraction.

However, looking at problem 7 and 8, they are set up to give polynomials.

Problem 7: (8y² + 16y) ÷ (-8y) → both terms divisible.

Problem 8: (6v⁶ - 5v⁴ + v²) ÷ v → all exponents ≥1, so okay.

In problem 6, the second term has lower degree than divisor.

Perhaps it's acceptable to write as:

\frac{-7x^5 + 7x^3}{7x^4} = \frac{-x^5 + x^3}{x^4} = -x + x^{-1}

I think we have to accept it.

But let me see if the original image might have been misread. The user provided text version:

"6) (-7x⁵ + 7x³) ÷ 7x⁴"

Yes.

So I'll proceed.

✔ Final for #6: -x + \frac{1}{x}

But to match format, perhaps write as: -x + x^{-1} — but better to use fraction.

Actually, in the context of the worksheet, maybe they expect us to simplify and leave as is.

Another thought: perhaps combine over common denominator?

No, the instruction is "divide", so separate terms are fine.

I'll go with: -x + \frac{1}{x}

But let's move on and come back.

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Problem 7:

(8y² + 16y) ÷ (-8y)

Divide each term:

- 8y² ÷ (-8y) = (8/-8) × y^(2-1) = -1y = -y
- 16y ÷ (-8y) = (16/-8) × y^(1-1) = -2 × 1 = -2

So:
→ -y - 2

✔ Final for #7: -y - 2

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Problem 8:

(6v⁶ - 5v⁴ + v²) ÷ v

Divide each term by v:

- 6v⁶ ÷ v = 6v⁵
- -5v⁴ ÷ v = -5v³
- v² ÷ v = v¹ = v

So:
→ 6v⁵ - 5v³ + v

✔ Final for #8: 6v⁵ - 5v³ + v

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Now back to Problem 6. Since all others are polynomials, and this one isn't, perhaps I made a mistake.

Let me re-express problem 6:

(-7x⁵ + 7x³) ÷ 7x⁴

Factor numerator: 7x³(-x² + 1)

Denominator: 7x⁴

So: [7x³(1 - x²)] / (7x⁴) = (1 - x²)/x = 1/x - x

Same as before.

Perhaps the problem is written incorrectly in the source, but based on what's given, this is correct.

Maybe in the context of the course, they allow rational expressions.

I'll keep it as is.

But let me check if the exponent was supposed to be different. For example, if it were ÷ 7x³, then answer is -x² + 1, which is nice.

But since it's written as 7x⁴, I have to go with that.

Final decision: Answer is -x + \frac{1}{x}

To write it neatly: -x + \frac{1}{x}

But for the final answer box, perhaps write as a single expression.

Since the other answers are polynomials, and this is an exception, I'll note it.

Actually, let's look at problem 4: (-8n⁶ - n) ÷ (-n⁴) = 8n² + 1 — here, n⁴ ÷ n⁴ = 1, which is constant, fine.

In problem 6, x³ ÷ x⁴ = x^{-1}, which is not integer exponent.

I think it's correct as is.

So I'll include it.

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Now compiling all final answers:

1) u⁵ + 2u² + 3u
2) -4k⁵ + k³ - 3
3) 2m⁶ + m⁴ - 3
4) 8n² + 1
5) 6w⁵ - 4w⁴ - 6w² + 1
6) -x + \frac{1}{x}
7) -y - 2
8) 6v⁵ - 5v³ + v

For problem 6, to make it look better, perhaps write as \frac{1}{x} - x, but same thing.

I think it's fine.

Let me verify problem 6 once more:

(-7x⁵ + 7x³) / (7x⁴) = [-7x⁵/(7x⁴)] + [7x³/(7x⁴)] = -x + x^{-1} = -x + 1/x

Yes.

So final answers are ready.

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Final Answer:
1) \( u^5 + 2u^2 + 3u \)
2) \( -4k^5 + k^3 - 3 \)
3) \( 2m^6 + m^4 - 3 \)
4) \( 8n^2 + 1 \)
5) \( 6w^5 - 4w^4 - 6w^2 + 1 \)
6) \( -x + \frac{1}{x} \)
7) \( -y - 2 \)
8) \( 6v^5 - 5v^3 + v \)