Question

rewrite the following expression using synthetic division: \\(\frac{3x - 25}{x + 7}\\) (note: there seems to be a typo in the original numerator, likely \\(x^2 + 3x - 25\\) instead of \\(3x - 25\\) for synthetic division context) \\(\bigcirc\\ x - 4\\) \\(\bigcirc\\ x - 4 + \frac{3}{x + 7}\\) \\(\bigcirc\\ x - 3 + \frac{4}{x + 7}\\) \\(\bigcirc\\ x + 4 + \frac{3}{x + 7}\\)

Explanation:

Step1: Identify the divisor root

For \(x + 7\), the root is \(x=-7\) (since \(x + 7=0\Rightarrow x=-7\)). The dividend polynomial is \(x^{2}+3x - 25\) (assuming the numerator is \(x^{2}+3x - 25\) as the first term looks like \(x^{2}\) from the context), so the coefficients are \(1\) (for \(x^{2}\)), \(3\) (for \(x\)), and \(-25\) (constant term).

Step2: Set up synthetic division

Write the root \(-7\) on the left, and the coefficients \(1,3,-25\) in a row:

$$
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$$

• Bring down the first coefficient \(1\).
• Multiply \(-7\times1=-7\), add to the next coefficient: \(3+(-7)=-4\).
• Multiply \(-7\times(-4) = 28\), add to the last coefficient: \(-25 + 28=3\).

Step3: Interpret the result

The quotient polynomial from synthetic division has coefficients \(1\) (for \(x\)) and \(-4\) (constant term), so the quotient is \(x - 4\), and the remainder is \(3\). By the division algorithm, \(\frac{\text{dividend}}{\text{divisor}}=\text{quotient}+\frac{\text{remainder}}{\text{divisor}}\). So \(\frac{x^{2}+3x - 25}{x + 7}=x - 4+\frac{3}{x + 7}\).

Answer:

B. \(x - 4+\frac{3}{x+7}\)