Polynomial review worksheet with algebra problems for practice.

Let’s solve each problem step by step. We’ll go one at a time, carefully.

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Problem 1: (3x⁻²)(2x⁴)

Multiply coefficients: 3 × 2 = 6
Add exponents of x: -2 + 4 = 2
→ 6x²

✔ Final Answer for #1: 6x²

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Problem 2: (5x³y⁻¹)/(10x⁻²y³)

Simplify coefficients: 5/10 = 1/2
For x: 3 - (-2) = 3 + 2 = 5 → x⁵
For y: -1 - 3 = -4 → y⁻⁴
So we have: (1/2)xy⁻⁴
But we must write with positive exponents only → move y⁴ to denominator
→ x⁵ / (2y⁴)

✔ Final Answer for #2: x⁵/(2y⁴)

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Problem 3: (4a²b⁻³c⁰)/(8a⁻¹b²c⁻²)

First, c⁰ = 1 → ignore it
Coefficients: 4/8 = 1/2
a: 2 - (-1) = 3 → a³
b: -3 - 2 = -5 → b⁻⁵ → move to denominator
c: 0 - (-2) = 2 → c² in numerator
So: (1/2) * a³ * c² / b⁵
→ a³c²/(2b)

✔ Final Answer for #3: a³c²/(2b⁵)

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Problem 4: (5x + x³ + 2x² - 15x + 6) + (2x³ - 7x² + 9x - 1)

Combine like terms:

- x⁴: 5x⁴
- x³: x³ + 2x³ = 3x³
- x²: 2x² - 7x² = -5x²
- x: -15x + 9x = -6x
- constants: 6 - 1 = 5

→ 5x⁴ + 3x³ - 5x² - 6x + 5

✔ Final Answer for #4: 5x⁴ + 3x³ - 5x² - 6x + 5

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Problem 5: (½m⁴ - 3mn + m²) - (¾m⁴ + mn - 3m²)

Distribute the minus sign:

= ½m⁴ - 3mn + m² - ¾m⁴ - mn + 3m²

Now combine:

m⁴: ½ - ¾ = -¼ → -¼m⁴
mn: -3mn - mn = -4mn
m²: m² + 3m² = 4m²

→ -¼m⁴ - 4mn + 4m²

We can rearrange in standard order (descending powers):

→ -¼m⁴ + 4m² - 4mn

✔ Final Answer for #5: -¼m⁴ + 4m² - 4mn

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Problem 6: (x² - 2x)(x³ - 3x² + 4)

Use distributive property (FOIL-style but for binomial × trinomial):

First: x²(x³ - 3x² + 4) = x⁵ - 3x⁴ + 4x²
Then: -2x(x³ - 3x² + 4) = -2x⁴ + 6x³ - 8x

Now add them together:

x⁵
-3x - 2x⁴ = -5x⁴
+6x³
+4x²
-8x

No constant term.

→ x⁵ - 5x⁴ + 6x³ + 4x² - 8x

✔ Final Answer for #6: x⁵ - 5x⁴ + 6x³ + 4x² - 8x

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Problem 7: (x² + 3x - 1)(x² - 2x + 5)

Multiply each term:

Start with x²(x² - 2x + 5) = x⁴ - 2x³ + 5x²
Then 3x(x² - 2x + 5) = 3x³ - 6x² + 15x
Then -1(x² - 2x + 5) = -x² + 2x - 5

Now add all:

x⁴
-2x³ + 3x³ = x³
5x² - 6x² - x² = -2x²
15x + 2x = 17x
-5

→ x + x³ - 2x² + 17x - 5

✔ Final Answer for #7: x⁴ + x³ - 2x² + 17x - 5

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Problem 8: (y + 6)(y - 6)

This is difference of squares: (a+b)(a-b) = a² - b²

Here, a = y, b = 6 → y² - 36

✔ Final Answer for #8: y² - 36

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Problem 9: (3x + 4)(x - 2)

Use FOIL:

First: 3x·x = 3x²
Outer: 3x·(-2) = -6x
Inner: 4·x = 4x
Last: 4·(-2) = -8

Combine: 3x² -6x +4x -8 = 3x² -2x -8

✔ Final Answer for #9: 3x² - 2x - 8

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Problem 10: (2y - 5)(3y + 4)

FOIL:

First: 2y·3y = 6y²
Outer: 2y·4 = 8y
Inner: -5·3y = -15y
Last: -5·4 = -20

Combine: 6y² +8y -15y -20 = 6y² -7y -20

✔ Final Answer for #10: 6y² - 7y - 20

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Problem 11: (2m - n)(2m + n)

Difference of squares again: (a-b)(a+b) = a² - b²

a = 2m, b = n → (2m)² - n² = 4m² - n²

✔ Final Answer for #11: 4m² - n²

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Problem 12: (3 - 2x)(2x - 4)

Let’s multiply directly:

First: 3·2x = 6x
Outer: 3·(-4) = -12
Inner: -2x·2x = -4x²
Last: -2x·(-4) = 8x

Now combine:

-4x²
6x + 8x = 14x
-12

→ -4x² + 14x - 12

We can factor out -2 if needed, but not required unless specified.

✔ Final Answer for #12: -4x² + 14x - 12

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Problem 13: (x² - 2x + 1)(x² - 1)

Notice: x² - 2x + 1 = (x - 1)²
And x² - 1 = (x - 1)(x + 1)

So overall: (x - 1)² · (x - 1)(x + 1) = (x - 1)³(x + 1)

But let’s expand fully as requested.

Multiply:

x²(x² - 1) = x⁴ - x²
-2x(x² - 1) = -2x³ + 2x
1(x² - 1) = x² - 1

Now add:

x⁴
-2x³
-x² + x² = 0
+2x
-1

→ x⁴ - 2x³ + 2x - 1

Wait — check that again:

From above:

x⁴ (from first)
-2x³ (from second)
-x² (first) + x² (third) = 0
+2x (second)
-1 (third)

Yes → x⁴ - 2x³ + 2x - 1

Alternatively, you could do polynomial multiplication vertically or use substitution, but this is correct.

✔ Final Answer for #13: x⁴ - 2x³ + 2x - 1

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## ✔ FINAL ANSWERS SUMMARY:

1. 6x²
2. x⁵/(2y⁴)
3. a³c²/(2b⁵)
4. 5x⁴ + 3x³ - 5x² - 6x + 5
5. -¼m⁴ + 4m² - 4mn
6. x⁵ - 5x⁴ + 6x³ + 4x² - 8x
7. x⁴ + x³ - 2x² + 17x - 5
8. y² - 36
9. 3x² - 2x - 8
10. 6y² - 7y - 20
11. 4m² - n²
12. -4x² + 14x - 12
13. x⁴ - 2x³ + 2x - 1

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Final Answer:
1. 6x²
2. x⁵/(2y⁴)
3. a³c²/(2b⁵)
4. 5x⁴ + 3x³ - 5x² - 6x + 5
5. -¼m⁴ + 4m² - 4mn
6. x⁵ - 5x⁴ + 6x³ + 4x² - 8x
7. x⁴ + x³ - 2x² + 17x - 5
8. y² - 36
9. 3x² - 2x - 8
10. 6y² - 7y - 20
11. 4m² - n²
12. -4x² + 14x - 12
13. x - 2x³ + 2x - 1