Practice dividing polynomials with this quiz and worksheet from Study.com.

Let's solve each of these polynomial division problems step by step using long division.

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Problem 1:

Divide:
$$
(6x^3 - 7x^2 + 2) \div (3x + 1)
$$

#### Step-by-step Long Division:

We divide $6x^3 - 7x^2 + 0x + 2$ by $3x + 1$. (Note: We include a $0x$ term for clarity.)

1. Divide: $ \frac{6x^3}{3x} = 2x^2 $
2. Multiply: $2x^2(3x + 1) = 6x^3 + 2x^2$
3. Subtract: $(6x^3 - 7x^2) - (6x^3 + 2x^2) = -9x^2$
4. Bring down the next term: $-9x^2 + 0x$

5. Divide: $ \frac{-9x^2}{3x} = -3x $
6. Multiply: $-3x(3x + 1) = -9x^2 - 3x$
7. Subtract: $(-9x^2 + 0x) - (-9x^2 - 3x) = 3x$

8. Bring down the next term: $3x + 2$

9. Divide: $ \frac{3x}{3x} = 1 $
10. Multiply: $1(3x + 1) = 3x + 1$
11. Subtract: $(3x + 2) - (3x + 1) = 1$

So, the result is:
$$
2x^2 - 3x + 1 + \frac{1}{3x + 1}
$$

✔ Answer: Option D
$$
\boxed{2x^2 - 3x + 1 + \frac{1}{3x + 1}}
$$

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Problem 2:

Divide:
$$
(2x^3 - 13x^2 + 17x + 12) \div (2x + 1)
$$

Use long division:

1. Divide: $ \frac{2x^3}{2x} = x^2 $
2. Multiply: $x^2(2x + 1) = 2x^3 + x^2$
3. Subtract: $(2x^3 - 13x^2) - (2x^3 + x^2) = -14x^2$
4. Bring down $+17x$: $-14x^2 + 17x$

5. Divide: $ \frac{-14x^2}{2x} = -7x $
6. Multiply: $-7x(2x + 1) = -14x^2 - 7x$
7. Subtract: $(-14x^2 + 17x) - (-14x^2 - 7x) = 24x$

8. Bring down $+12$: $24x + 12$

9. Divide: $ \frac{24x}{2x} = 12 $
10. Multiply: $12(2x + 1) = 24x + 12$
11. Subtract: $(24x + 12) - (24x + 12) = 0$

No remainder!

So, the quotient is:
$$
x^2 - 7x + 12
$$

✔ Answer: Option A
$$
\boxed{x^2 - 7x + 12}
$$

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Problem 3:

Divide:
$$
(2x^3 - 5x^2 - 28x + 15) \div (x - 2)
$$

Use long division:

1. Divide: $ \frac{2x^3}{x} = 2x^2 $
2. Multiply: $2x^2(x - 2) = 2x^3 - 4x^2$
3. Subtract: $(2x^3 - 5x^2) - (2x^3 - 4x^2) = -x^2$
4. Bring down $-28x$: $-x^2 - 28x$

5. Divide: $ \frac{-x^2}{x} = -x $
6. Multiply: $-x(x - 2) = -x^2 + 2x$
7. Subtract: $(-x^2 - 28x) - (-x^2 + 2x) = -30x$

8. Bring down $+15$: $-30x + 15$

9. Divide: $ \frac{-30x}{x} = -30 $
10. Multiply: $-30(x - 2) = -30x + 60$
11. Subtract: $(-30x + 15) - (-30x + 60) = -45$

So, the result is:
$$
2x^2 - x - 30 + \frac{-45}{x - 2}
$$

✔ Answer: Option A
$$
\boxed{2x^2 - x - 30 + \frac{-45}{x - 2}}
$$

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✔ Final Answers:

1. $ \boxed{2x^2 - 3x + 1 + \frac{1}{3x + 1}} $
2. $ \boxed{x^2 - 7x + 12} $
3. $ \boxed{2x^2 - x - 30 + \frac{-45}{x - 2}} $

Let me know if you'd like to see synthetic division for any of these!