Question

1. divide using long division.

$(2x^3 - 5x^2 - 4x - 25) \div (x - 4)$
options:
$2x^3 + 3x + \frac{7}{x - 4}$
$2x^2 - 13x + 48 + \frac{217}{x - 4}$
$2x^2 - x + 10 + \frac{7}{x - 4}$
$2x^2 + 3x + 8 + \frac{7}{x - 4}$

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(2x^3 - 5x^2 - 4x - 25\) by the leading term of the divisor \(x - 4\). So, \(\frac{2x^3}{x}=2x^2\).

Step2: Multiply and subtract

Multiply the divisor \(x - 4\) by \(2x^2\) to get \(2x^3 - 8x^2\). Subtract this from the dividend:

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Step3: Divide the new leading term

Divide the leading term of \(3x^2 - 4x - 25\) (which is \(3x^2\)) by the leading term of \(x - 4\) (which is \(x\)) to get \(3x\).

Step4: Multiply and subtract

Multiply the divisor \(x - 4\) by \(3x\) to get \(3x^2 - 12x\). Subtract this from \(3x^2 - 4x - 25\):

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Step5: Divide the new leading term

Divide the leading term of \(8x - 25\) (which is \(8x\)) by the leading term of \(x - 4\) (which is \(x\)) to get \(8\).

Step6: Multiply and subtract

Multiply the divisor \(x - 4\) by \(8\) to get \(8x - 32\). Subtract this from \(8x - 25\):

$$
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$$

Step7: Write the final result

The quotient is \(2x^2 + 3x + 8\) and the remainder is \(7\), so the result of the division \((2x^3 - 5x^2 - 4x - 25)\div(x - 4)\) is \(2x^2 + 3x + 8+\frac{7}{x - 4}\).

Answer:

\(2x^2 + 3x + 8+\frac{7}{x - 4}\) (which corresponds to the first option among the given choices: \(2x^2 + 3x + 8+\frac{7}{x - 4}\))