Question

1. use a computer algebra system to find the quotient for the division problem below.

divide $2x^3 + x^2 + 5x - 3$ by $x - 1$.
\\(\bigcirc\\) $x^2 + 4x + 6 + \dfrac{2}{x - 1}$
\\(\bigcirc\\) $4x^2 + 6x + 4 + \dfrac{2}{x - 1}$
\\(\bigcirc\\) $2x^2 + 3x + 8 + \dfrac{5}{x - 1}$
\\(\bigcirc\\) $2x^2 + 7x - 12 + \dfrac{5}{x - 1}$

Explanation:

Step 1: Polynomial Long Division Setup

We divide the polynomial \(2x^{3}+x^{2}+5x - 3\) by \(x - 1\). Using polynomial long division, we first divide the leading term of the dividend \(2x^{3}\) by the leading term of the divisor \(x\), which gives \(2x^{2}\).

Step 2: Multiply and Subtract

Multiply the divisor \(x - 1\) by \(2x^{2}\) to get \(2x^{3}-2x^{2}\). Subtract this from the dividend: \((2x^{3}+x^{2}+5x - 3)-(2x^{3}-2x^{2})=3x^{2}+5x - 3\).

Step 3: Repeat the Process

Now, divide the leading term of \(3x^{2}+5x - 3\) (which is \(3x^{2}\)) by the leading term of the divisor \(x\) to get \(3x\). Multiply the divisor \(x - 1\) by \(3x\) to get \(3x^{2}-3x\). Subtract this from \(3x^{2}+5x - 3\): \((3x^{2}+5x - 3)-(3x^{2}-3x)=8x - 3\).

Step 4: Repeat Again

Divide the leading term of \(8x - 3\) (which is \(8x\)) by the leading term of the divisor \(x\) to get \(8\). Multiply the divisor \(x - 1\) by \(8\) to get \(8x - 8\). Subtract this from \(8x - 3\): \((8x - 3)-(8x - 8)=5\).

Step 5: Write the Result

So, the division gives us \(2x^{2}+3x + 8+\frac{5}{x - 1}\), which is the third option.

Answer:

\(2x^{2}+3x + 8+\frac{5}{x - 1}\) (The third option: \(2x^{2}+3x+8+\frac{5}{x - 1}\))