Question

introduction homework (standard 0)
score: 3/28 answered: 3/25
question 4
find the quotient and remainder using long division.
\\(\frac{x^4 - x^3 + 4x^2 + x + 1}{x - 1}\\)
the quotient is
the remainder is
question help: video 1 video 2 message instructor post to forum

Explanation:

Step1: Divide the leading term

Divide the leading term of the dividend \(x^4 - x^3 + 4x^2 + x + 1\) (which is \(x^4\)) by the leading term of the divisor \(x - 1\) (which is \(x\)). We get \(x^3\).
Multiply the divisor \(x - 1\) by \(x^3\): \(x^3(x - 1)=x^4 - x^3\).
Subtract this from the dividend: \((x^4 - x^3 + 4x^2 + x + 1)-(x^4 - x^3)=4x^2 + x + 1\).

Step2: Divide the new leading term

Now, take the leading term of the new dividend \(4x^2\) and divide by \(x\) (the leading term of the divisor). We get \(4x\).
Multiply the divisor \(x - 1\) by \(4x\): \(4x(x - 1)=4x^2 - 4x\).
Subtract this from the new dividend: \((4x^2 + x + 1)-(4x^2 - 4x)=5x + 1\).

Step3: Divide the new leading term again

Take the leading term of the new dividend \(5x\) and divide by \(x\). We get \(5\).
Multiply the divisor \(x - 1\) by \(5\): \(5(x - 1)=5x - 5\).
Subtract this from the new dividend: \((5x + 1)-(5x - 5)=6\).

Step4: Determine quotient and remainder

The quotient is the sum of the terms we got in each division step: \(x^3 + 4x + 5\).
The remainder is the final result after the last subtraction, which is \(6\).

Answer:

The quotient is \(x^3 + 4x + 5\) and the remainder is \(6\).