Question

introduction homework (standard 0)
score: 5/28 answered: 5/25
question 6
find the quotient and remainder using long division.
\\(\frac{x^3 - 15x}{x - 4}\\)
the quotient is
the remainder is
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Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(x^3 - 15x\) (which is \(x^3\)) by the leading term of the divisor \(x - 4\) (which is \(x\)). So, \( \frac{x^3}{x}=x^2\).
Multiply the divisor \(x - 4\) by \(x^2\): \(x^2(x - 4)=x^3 - 4x^2\).
Subtract this from the dividend: \((x^3 - 15x)-(x^3 - 4x^2)=4x^2 - 15x\).

Step2: Divide the new leading term

Now, take the leading term of \(4x^2 - 15x\) (which is \(4x^2\)) and divide by the leading term of the divisor \(x\). So, \( \frac{4x^2}{x}=4x\).
Multiply the divisor \(x - 4\) by \(4x\): \(4x(x - 4)=4x^2 - 16x\).
Subtract this from \(4x^2 - 15x\): \((4x^2 - 15x)-(4x^2 - 16x)=x\).

Step3: Divide the new leading term

Take the leading term of \(x\) (which is \(x\)) and divide by the leading term of the divisor \(x\). So, \( \frac{x}{x}=1\).
Multiply the divisor \(x - 4\) by \(1\): \(1\times(x - 4)=x - 4\).
Subtract this from \(x\): \(x-(x - 4)=4\).

Step4: Determine quotient and remainder

The quotient is the sum of the terms we found in each division step: \(x^2 + 4x + 1\).
The remainder is the final result of the subtraction, which is \(4\).

Answer:

The quotient is \(x^2 + 4x + 1\)
The remainder is \(4\)