Question

find the quotient.
\\((x^4 + 3x^3 + 2x^2 + 12x - 8) \div (x + 3)\\)

a. \\(x^3 + 2x + 6 - \frac{26}{x + 3}\\)

b. \\(x^3 + 2x + 6 + \frac{10}{x + 3}\\)

c. \\(x^3 + 2x - 6 + \frac{10}{x + 3}\\)

d. \\(x^3 + 2x^2 + 6x - \frac{26}{x + 3}\\)

Explanation:

Based on the image provided, the topic is Synthetic Division with Remainders.

The problem asks to find the quotient for:

$$ (x^4 + 3x^3 + 3x^2 + 12x - 8) \div (x + 3) $$

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🆕 New Concept Discovered: Synthetic Division
Using a simplified shorthand method to divide polynomials.

Step 1: Set up the synthetic division

To divide by \(x + 3\), we use the root \(c = -3\). We write down the coefficients of the dividend polynomial \(x^4 + 3x^3 + 3x^2 + 12x - 8\):

$$ [1, \quad 3, \quad 3, \quad 12, \quad -8] $$

Set up the division table:

$$
LATEXBLOCK0
$$

Step 2: Perform the synthetic division steps

1. Bring down the first coefficient, \(1\):

$$
LATEXBLOCK1
$$

1. Multiply \(1\) by \(-3\) to get \(-3\), place it under the next coefficient, and add:

$$
LATEXBLOCK2
$$

1. Multiply \(0\) by \(-3\) to get \(0\), place it under the next coefficient, and add:

$$
LATEXBLOCK3
$$

1. Multiply \(3\) by \(-3\) to get \(-9\), place it under the next coefficient, and add:

$$
LATEXBLOCK4
$$

1. Multiply \(3\) by \(-3\) to get \(-9\), place it under the last coefficient, and add:

$$
LATEXBLOCK5
$$

Step 3: Write the resulting polynomial

The bottom row values \([1, 0, 3, 3, -17]\) represent the coefficients of the quotient polynomial (which starts one degree lower than the original, so \(x^3\)) and the final number is the remainder:

• Coefficient of \(x^3\): \(1\)
• Coefficient of \(x^2\): \(0\)
• Coefficient of \(x\): \(3\)
• Constant term: \(3\)
• Remainder: \(-17\)

This gives:

$$ x^3 + 3x + 3 - \frac{17}{x + 3} $$

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Answer:

$$ x^3 + 3x + 3 - \frac{17}{x + 3} $$