Polynomials Worksheets with Answer Key - Free Printable

Let’s go through each problem one by one, carefully performing the indicated operations.

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Problem 1: (8b - 2)(3b² + 2b - 2)
We use distribution (FOIL or box method). Multiply each term in the first polynomial by each term in the second.

First: 8b × 3b² = 24b³
8b × 2b = 16b²
8b × (-2) = -16b

Then: -2 × 3b² = -6b²
-2 × 2b = -4b
-2 × (-2) = +4

Now combine like terms:
24b³ + (16b² - 6b²) + (-16b - 4b) + 4
= 24b³ + 10b² - 20b + 4

✔ Final for #1: 24b³ + 10b² - 20b + 4

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Problem 2: (3x² - 2x + 1) + (-x² + 3x + 1)
Add like terms:

x² terms: 3x² + (-x²) = 2x²
x terms: -2x + 3x = x
constants: 1 + 1 = 2

✔ Final for #2: 2x² + x + 2

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Problem 3: (-3m² + m) + (4m² + 6m)
Combine like terms:

m²: -3m² + 4m² = m²
m: m + 6m = 7m

✔ Final for #3: m² + 7m

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Problem 4: (7a² - a + 4) - (3a² - 4a - 3)
Distribute the minus sign to the second polynomial:

7a² - a + 4 - 3a² + 4a + 3

Now combine:

a²: 7a² - 3a² = 4a²
a: -a + 4a = 3a
constants: 4 + 3 = 7

✔ Final for #4: 4a² + 3a + 7

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Problem 5: (-3x² + 6x³ - 4 - x) ÷ (2x + 1)
First, rewrite dividend in standard order: 6x³ - 3x² - x - 4

Use polynomial long division:

Divide 6x³ by 2x → 3x²
Multiply 3x²(2x+1) = 6x³ + 3x²
Subtract: (6x³ - 3x²) - (6x³ + 3x²) = -6x²
Bring down next term: -6x² - x

Divide -6x² by 2x → -3x
Multiply -3x(2x+1) = -6x² - 3x
Subtract: (-6x² - x) - (-6x² - 3x) = 2x
Bring down -4 → 2x - 4

Divide 2x by 2x → 1
Multiply 1(2x+1) = 2x + 1
Subtract: (2x - 4) - (2x + 1) = -5

So quotient is 3x² - 3x + 1, remainder -5

Answer: 3x² - 3x + 1 - 5/(2x+1)
But since it says “perform the operation”, and no specification, we can write as:

✔ Final for #5: 3x² - 3x + 1 R -5 or 3x² - 3x + 1 - \frac{5}{2x+1}

But in many school contexts, they expect the quotient with remainder notation. Let’s stick with:

3x² - 3x + 1 with remainder -5

Actually, let me double-check the subtraction steps:

After first step:
6x³ - 3x² - x - 4
minus (6x³ + 3x²) → gives -6x² - x - 4? Wait — I think I missed bringing down the constant early.

Let me redo carefully:

Dividend: 6x³ - 3x² - x - 4
Divisor: 2x + 1

Step 1: 6x³ ÷ 2x = 3x²
3x²*(2x+1) = 6x³ + 3x²
Subtract: (6x³ - 3x²) - (6x³ + 3x²) = -6x²
Bring down -x → now have -6x² - x

Step 2: -6x² ÷ 2x = -3x
-3x*(2x+1) = -6x² - 3x
Subtract: (-6x² - x) - (-6x² - 3x) = 2x
Bring down -4 → 2x - 4

Step 3: 2x ÷ 2x = 1
1*(2x+1) = 2x + 1
Subtract: (2x - 4) - (2x + 1) = -5

Yes, correct.

So answer is: 3x² - 3x + 1 with remainder -5

In fractional form: 3x² - 3x + 1 - \frac{5}{2x+1}

But since the problem doesn’t specify format, and others are simplifications, perhaps leave as quotient and remainder.

However, looking at other problems, they’re all simplifying expressions, not leaving remainders. But this is division — so we must include remainder.

I’ll write: 3x² - 3x + 1 - \frac{5}{2x+1}

But let’s check if it divides evenly? No, remainder is -5.

Alternatively, maybe the problem expects us to arrange terms properly and divide.

Wait — original was written as: (-3x² + 6x³ - 4 - x) — which is same as 6x³ - 3x² - x - 4. Correct.

Okay, final answer for #5: 3x² - 3x + 1 - \frac{5}{2x+1}

But in some curricula, they write it as:

\frac{6x^3 - 3x^2 - x - 4}{2x + 1} = 3x^2 - 3x + 1 - \frac{5}{2x+1}

Since the instruction is “perform the indicated operation”, and it’s division, this is acceptable.

But let’s see if we can factor or something? Probably not necessary.

I’ll go with the quotient and remainder expression.

Actually, upon second thought — in many middle/high school worksheets, for polynomial division, they accept the answer as:

3x² - 3x + 1 R -5

But to be precise, I’ll write it as a mixed expression.

Final decision: 3x² - 3x + 1 - \frac{5}{2x+1}

But let’s move on and come back if needed.

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Problem 6: 2a(5a² + 8a + 8)
Distribute 2a:

2a * 5a² = 10a³
2a * 8a = 16a²
2a * 8 = 16a

✔ Final for #6: 10a³ + 16a² + 16a

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Problem 7: (-18p² + p - 32) - (40 - 13p²)
Distribute the minus:

-18p² + p - 32 - 40 + 13p²

Combine:

p²: -18p² + 13p² = -5p²
p: p
constants: -32 - 40 = -72

✔ Final for #7: -5p² + p - 72

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Problem 8: -8w²y + (4w²y⁴ - w⁴)
This is addition. Just remove parentheses:

-8w²y + 4w²y⁴ - w⁴

No like terms to combine (different powers).

Order by degree or alphabetically? Usually descending powers of y or w.

But since no instruction, leave as is.

We can write: 4w²y⁴ - 8w²y - w⁴

Or group w² terms: w²(4y⁴ - 8y) - w⁴ — but not necessary.

Best to write in standard form — perhaps by descending powers of y.

Terms: 4w²y⁴ (degree 6), -8w²y (degree 3), -w⁴ (degree 4) — messy.

Actually, variables are mixed. Best to leave as simplified sum.

So: 4w²y⁴ - 8w²y - w⁴

✔ Final for #8: 4w²y⁴ - 8w²y - w⁴

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Problem 9: (x - 2)(x² - x + 3)
Multiply each term:

x * x² = x³
x * (-x) = -x²
x * 3 = 3x

-2 * x² = -2x²
-2 * (-x) = +2x
-2 * 3 = -6

Combine:

x³ + (-x² - 2x²) + (3x + 2x) - 6
= x³ - 3x² + 5x - 6

✔ Final for #9: x³ - 3x² + 5x - 6

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Problem 10: (5x³ - 13x² - 7) + (16x³ + 8x² - x + 15)
Combine like terms:

x³: 5x³ + 16x³ = 21x³
x²: -13x² + 8x² = -5x²
x: 0x - x = -x
constants: -7 + 15 = 8

✔ Final for #10: 21x³ - 5x² - x + 8

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Now, compiling all answers:

1. 24b³ + 10b² - 20b + 4
2. 2x² + x + 2
3. m² + 7m
4. 4a² + 3a + 7
5. 3x² - 3x + 1 - \frac{5}{2x+1} (or with remainder)
6. 10a³ + 16a² + 16a
7. -5p² + p - 72
8. 4w²y⁴ - 8w²y - w⁴
9. x³ - 3x² + 5x - 6
10. 21x³ - 5x² - x + 8

For #5, since it's division and remainder is non-zero, and the problem says "perform the operation", I think writing the quotient and remainder is fine. But to match the style of other answers (which are polynomials), perhaps they expect the division result as a polynomial plus remainder fraction.

I'll keep it as: 3x² - 3x + 1 - \frac{5}{2x+1}

But let me verify the division once more quickly:

(2x+1)(3x² - 3x + 1) = 2x(3x² - 3x + 1) + 1(3x² - 3x + 1) = 6x³ - 6x² + 2x + 3x² - 3x + 1 = 6x³ - 3x² - x + 1

But our dividend is 6x³ - 3x² - x - 4

So difference: (6x³ - 3x² - x - 4) - (6x³ - 3x² - x + 1) = -5

Yes, so (2x+1)(3x² - 3x + 1) - 5 = dividend

Thus, dividend / divisor = 3x² - 3x + 1 - 5/(2x+1)

Correct.

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Final Answers:

1. 24b³ + 10b² - 20b + 4
2. 2x² + x + 2
3. m² + 7m
4. 4a² + 3a + 7
5. 3x² - 3x + 1 - \frac{5}{2x+1}
6. 10a³ + 16a² + 16a
7. -5p² + p - 72
8. 4w²y⁴ - 8w²y - w⁴
9. x³ - 3x² + 5x - 6
10. 21x³ - 5x² - x + 8

Note: For #5, if the context expects only polynomial quotient, it might be listed as 3x² - 3x + 1 with remainder -5, but since the operation is division, including the fractional part is more complete.

I'll present them as above.

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Final Answer:
1. \( 24b^3 + 10b^2 - 20b + 4 \)
2. \( 2x^2 + x + 2 \)
3. \( m^2 + 7m \)
4. \( 4a^2 + 3a + 7 \)
5. \( 3x^2 - 3x + 1 - \frac{5}{2x+1} \)
6. \( 10a^3 + 16a^2 + 16a \)
7. \( -5p^2 + p - 72 \)
8. \( 4w^2y^4 - 8w^2y - w^4 \)
9. \( x^3 - 3x^2 + 5x - 6 \)
10. \( 21x^3 - 5x^2 - x + 8 \)