Question

1. use a computer algebra system to find the quotient for the division problem below. divide 2x^3 + x^2 + 5x - 3 by x - 1. x^2+4x + 6+\frac{2}{x - 1} 2x^2+3x + 8+\frac{5}{x - 1} 2x^2+7x - 12+\frac{5}{x - 1} 4x^2+6x + 4+\frac{2}{x - 1}

Explanation:

Step1: Use polynomial long - division

We divide \(2x^{3}+x^{2}+5x - 3\) by \(x - 1\). First, divide the leading term of the dividend \(2x^{3}\) by the leading term of the divisor \(x\), we get \(2x^{2}\). Then multiply \(x - 1\) by \(2x^{2}\) to get \(2x^{3}-2x^{2}\), and subtract it from the dividend: \((2x^{3}+x^{2}+5x - 3)-(2x^{3}-2x^{2})=3x^{2}+5x - 3\).

Step2: Repeat the process

Divide the leading term of the new dividend \(3x^{2}\) by the leading term of the divisor \(x\) to get \(3x\). Multiply \(x - 1\) by \(3x\) to get \(3x^{2}-3x\), and subtract it from the new dividend: \((3x^{2}+5x - 3)-(3x^{2}-3x)=8x - 3\).

Step3: One more time

Divide the leading term of the new dividend \(8x\) by the leading term of the divisor \(x\) to get \(8\). Multiply \(x - 1\) by \(8\) to get \(8x-8\), and subtract it from the new dividend: \((8x - 3)-(8x - 8)=5\).
The quotient is \(2x^{2}+3x + 8\) and the remainder is \(5\), so the result is \(2x^{2}+3x + 8+\frac{5}{x - 1}\).

Answer:

\(2x^{2}+3x + 8+\frac{5}{x - 1}\) (corresponding to the second option in the multiple - choice)