Question

hw 3.5: polynomial division
score: 4/15 answered: 4/15
question 5
divide the following. give your answer in the form quotient + \\(\frac{\text{remainder}}{\text{divisor}}\\)
\\(\frac{x^3 + 5x^2 - 1}{x + 1} = \\)
question help: video message instructor
submit question jump to answer

Explanation:

Step1: Perform polynomial long division

Divide \(x^3 + 5x^2 - 1\) by \(x + 1\).
First term: \(\frac{x^3}{x}=x^2\). Multiply \(x + 1\) by \(x^2\) to get \(x^3 + x^2\). Subtract from \(x^3 + 5x^2 - 1\): \((x^3 + 5x^2 - 1)-(x^3 + x^2)=4x^2 - 1\).

Step2: Divide the new polynomial

Next term: \(\frac{4x^2}{x}=4x\). Multiply \(x + 1\) by \(4x\) to get \(4x^2 + 4x\). Subtract from \(4x^2 - 1\): \((4x^2 - 1)-(4x^2 + 4x)=-4x - 1\).

Step3: Divide the new polynomial

Next term: \(\frac{-4x}{x}=-4\). Multiply \(x + 1\) by \(-4\) to get \(-4x - 4\). Subtract from \(-4x - 1\): \((-4x - 1)-(-4x - 4)=3\).

Step4: Write the result

The quotient is \(x^2 + 4x - 4\) and the remainder is \(3\). So, \(\frac{x^3 + 5x^2 - 1}{x + 1}=x^2 + 4x - 4+\frac{3}{x + 1}\).

Answer:

\(x^2 + 4x - 4+\frac{3}{x + 1}\)