Question

divide using polynomial long division.

(3x² - 14x - 5) ÷ (x - 5) = □

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(3x^2\) by the leading term of the divisor \(x\). So, \(\frac{3x^2}{x} = 3x\). This is the first term of the quotient.

Step2: Multiply and subtract

Multiply the divisor \((x - 5)\) by \(3x\) to get \(3x(x - 5)=3x^2-15x\). Now, subtract this from the dividend:

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Step3: Divide the new leading terms

Now, divide the leading term of the new dividend \(x\) by the leading term of the divisor \(x\). So, \(\frac{x}{x}=1\). This is the next term of the quotient.

Step4: Multiply and subtract (final)

Multiply the divisor \((x - 5)\) by \(1\) to get \(1\times(x - 5)=x - 5\). Subtract this from the new dividend \(x - 5\):

$$ (x - 5)-(x - 5)=x - 5 - x + 5 = 0 $$

Since the remainder is \(0\), the quotient is \(3x + 1\).

Answer:

\(3x + 1\)