Polynomial long division of 27x³ + 9x² - 3x - 9 by 3x - 2, with step-by-step solution and helpful hints.

The image shows a polynomial long division problem where we are dividing:

$$
\frac{27x^3 + 9x^2 - 3x - 9}{3x - 2}
$$

Let’s go step by step and verify the solution, then explain it clearly.

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🔷 Step-by-Step Polynomial Division

We are dividing:
$$
\frac{27x^3 + 9x^2 - 3x - 9}{3x - 2}
$$

We'll use long division.

#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.

$$
\frac{27x^3}{3x} = 9x^2
$$

So, the first term of the quotient is $ 9x^2 $. Multiply this by the divisor:

$$
9x^2 \cdot (3x - 2) = 27x^3 - 18x^2
$$

Subtract this from the original polynomial:

$$
(27x^3 + 9x^2 - 3x - 9) - (27x^3 - 18x^2) = (0x^3) + (9x^2 + 18x^2) - 3x - 9 = 27x^2 - 3x - 9
$$

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#### Step 2: Repeat with the new polynomial: $ 27x^2 - 3x - 9 $

Divide leading terms:

$$
\frac{27x^2}{3x} = 9x
$$

Add $ 9x $ to the quotient. Multiply:

$$
9x \cdot (3x - 2) = 27x^2 - 18x
$$

Subtract:

$$
(27x^2 - 3x - 9) - (27x^2 - 18x) = (-3x + 18x) - 9 = 15x - 9
$$

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#### Step 3: Now divide $ 15x - 9 $

$$
\frac{15x}{3x} = 5
$$

Add $ 5 $ to the quotient. Multiply:

$$
5 \cdot (3x - 2) = 15x - 10
$$

Subtract:

$$
(15x - 9) - (15x - 10) = (-9 + 10) = 1
$$

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✔ Final Result:

- Quotient: $ 9x^2 + 9x + 5 $
- Remainder: $ 1 $

So, we can write:

$$
\frac{27x^3 + 9x^2 - 3x - 9}{3x - 2} = 9x^2 + 9x + 5 + \frac{1}{3x - 2}
$$

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📌 Explanation of Hints on the Right

The hints show how each term in the quotient was derived:

1. $ \frac{27x^3}{3x} = 9x^2 $ → First term.
2. $ 9x^2(3x - 2) = 27x^3 - 18x^2 $ → Used for subtraction.
3. $ \frac{27x^2}{3x} = 9x $ → Second term.
4. $ 9x(3x - 2) = 27x^2 - 18x $ → Next subtraction.
5. $ \frac{15x}{3x} = 5 $ → Third term.
6. $ 5(3x - 2) = 15x - 10 $ → Final multiplication.

Then, subtracting gives remainder $ 1 $, which cannot be divided further.

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✔ Final Answer:

$$
\boxed{9x^2 + 9x + 5 + \frac{1}{3x - 2}}
$$

This is the complete result of the division. The process shown in the image is correct!