Algebra worksheet featuring polynomial long division exercises for practice.

1. $3x^2 + 12x + 10$ divided by $x + 2$
Using polynomial long division:
- Divide $3x^2$ by $x$ to get $3x$.
- Multiply $3x$ by $x + 2$ to get $3x^2 + 6x$.
- Subtract: $(3x^2 + 12x + 10) - (3x^2 + 6x) = 6x + 10$.
- Divide $6x$ by $x$ to get $6$.
- Multiply $6$ by $x + 2$ to get $6x + 12$.
- Subtract: $(6x + 10) - (6x + 12) = -2$.
Answer: $3x + 6 - \frac{2}{x + 2}$

2. $2x^3 - x^2 - 30x + 50$ divided by $x - 3$
Using polynomial long division:
- Divide $2x^3$ by $x$ to get $2x^2$.
- Multiply $2x^2$ by $x - 3$ to get $2x^3 - 6x^2$.
- Subtract: $(2x^3 - x^2 - 30x + 50) - (2x^3 - 6x^2) = 5x^2 - 30x + 50$.
- Divide $5x^2$ by $x$ to get $5x$.
- Multiply $5x$ by $x - 3$ to get $5x^2 - 15x$.
- Subtract: $(5x^2 - 30x + 50) - (5x^2 - 15x) = -15x + 50$.
- Divide $-15x$ by $x$ to get $-15$.
- Multiply $-15$ by $x - 3$ to get $-15x + 45$.
- Subtract: $(-15x + 50) - (-15x + 45) = 5$.
Answer: $2x^2 + 5x - 15 + \frac{5}{x - 3}$

3. $3x^3 - 12x + 24$ divided by $x - 2$
Using polynomial long division:
- Divide $3x^3$ by $x$ to get $3x^2$.
- Multiply $3x^2$ by $x - 2$ to get $3x^3 - 6x^2$.
- Subtract: $(3x^3 - 12x + 24) - (3x^3 - 6x^2) = 6x^2 - 12x + 24$.
- Divide $6x^2$ by $x$ to get $6x$.
- Multiply $6x$ by $x - 2$ to get $6x^2 - 12x$.
- Subtract: $(6x^2 - 12x + 24) - (6x^2 - 12x) = 24$.
- Divide $24$ by $x$ to get $0$.
Answer: $3x^2 + 6x + \frac{24}{x - 2}$

4. $4x^3 + 5x^2 + 5x - 5$ divided by $x + 1$
Using polynomial long division:
- Divide $4x^3$ by $x$ to get $4x^2$.
- Multiply $4x^2$ by $x + 1$ to get $4x^3 + 4x^2$.
- Subtract: $(4x^3 + 5x^2 + 5x - 5) - (4x^3 + 4x^2) = x^2 + 5x - 5$.
- Divide $x^2$ by $x$ to get $x$.
- Multiply $x$ by $x + 1$ to get $x^2 + x$.
- Subtract: $(x^2 + 5x - 5) - (x^2 + x) = 4x - 5$.
- Divide $4x$ by $x$ to get $4$.
- Multiply $4$ by $x + 1$ to get $4x + 4$.
- Subtract: $(4x - 5) - (4x + 4) = -9$.
Answer: $4x^2 + x + 4 - \frac{9}{x + 1}$

5. $4x^3 - 38x^2 + 22x + 2$ divided by $x - 9$
Using polynomial long division:
- Divide $4x^3$ by $x$ to get $4x^2$.
- Multiply $4x^2$ by $x - 9$ to get $4x^3 - 36x^2$.
- Subtract: $(4x^3 - 38x^2 + 22x + 2) - (4x^3 - 36x^2) = -2x^2 + 22x + 2$.
- Divide $-2x^2$ by $x$ to get $-2x$.
- Multiply $-2x$ by $x - 9$ to get $-2x^2 + 18x$.
- Subtract: $(-2x^2 + 22x + 2) - (-2x^2 + 18x) = 4x + 2$.
- Divide $4x$ by $x$ to get $4$.
- Multiply $4$ by $x - 9$ to get $4x - 36$.
- Subtract: $(4x + 2) - (4x - 36) = 38$.
Answer: $4x^2 - 2x + 4 + \frac{38}{x - 9}$

6. $4x^3 + 5x^2 - 8x + 30$ divided by $x + 3$
Using polynomial long division:
- Divide $4x^3$ by $x$ to get $4x^2$.
- Multiply $4x^2$ by $x + 3$ to get $4x^3 + 12x^2$.
- Subtract: $(4x^3 + 5x^2 - 8x + 30) - (4x^3 + 12x^2) = -7x^2 - 8x + 30$.
- Divide $-7x^2$ by $x$ to get $-7x$.
- Multiply $-7x$ by $x + 3$ to get $-7x^2 - 21x$.
- Subtract: $(-7x^2 - 8x + 30) - (-7x^2 - 21x) = 13x + 30$.
- Divide $13x$ by $x$ to get $13$.
- Multiply $13$ by $x + 3$ to get $13x + 39$.
- Subtract: $(13x + 30) - (13x + 39) = -9$.
Answer: $4x^2 - 7x + 13 - \frac{9}{x + 3}$

7. $2x^3 + 9x^2 + 14x + 10$ divided by $x + 3$
Using polynomial long division:
- Divide $2x^3$ by $x$ to get $2x^2$.
- Multiply $2x^2$ by $x + 3$ to get $2x^3 + 6x^2$.
- Subtract: $(2x^3 + 9x^2 + 14x + 10) - (2x^3 + 6x^2) = 3x^2 + 14x + 10$.
- Divide $3x^2$ by $x$ to get $3x$.
- Multiply $3x$ by $x + 3$ to get $3x^2 + 9x$.
- Subtract: $(3x^2 + 14x + 10) - (3x^2 + 9x) = 5x + 10$.
- Divide $5x$ by $x$ to get $5$.
- Multiply $5$ by $x + 3$ to get $5x + 15$.
- Subtract: $(5x + 10) - (5x + 15) = -5$.
Answer: $2x^2 + 3x + 5 - \frac{5}{x + 3}$

8. $2x^3 - 8x^2 - 4x^3 + 8x + 1$ divided by $x - 1$
First, simplify the polynomial:
$2x^3 - 4x^3 - 8x^2 + 8x + 1 = -2x^3 - 8x^2 + 8x + 1$
Now divide $-2x^3 - 8x^2 + 8x + 1$ by $x - 1$:
- Divide $-2x^3$ by $x$ to get $-2x^2$.
- Multiply $-2x^2$ by $x - 1$ to get $-2x^3 + 2x^2$.
- Subtract: $(-2x^3 - 8x^2 + 8x + 1) - (-2x^3 + 2x^2) = -10x^2 + 8x + 1$.
- Divide $-10x^2$ by $x$ to get $-10x$.
- Multiply $-10x$ by $x - 1$ to get $-10x^2 + 10x$.
- Subtract: $(-10x^2 + 8x + 1) - (-10x^2 + 10x) = -2x + 1$.
- Divide $-2x$ by $x$ to get $-2$.
- Multiply $-2$ by $x - 1$ to get $-2x + 2$.
- Subtract: $(-2x + 1) - (-2x + 2) = -1$.
Answer: $-2x^2 - 10x - 2 - \frac{1}{x - 1}$