What is the quotient of (x^3 - 3x^2 + 5x - 3) divided by (x - 1)?

1. Set up polynomial long division: We want to divide the polynomial $x^3 - 3x^2 + 5x - 3$ by the binomial $x - 1$. We set this up like a standard long division problem.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -2x & +3 \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ \end{array} $$

1. Divide the leading terms: Divide the leading term of the dividend ($x^3$) by the leading term of the divisor ($x$). This gives $x^3 / x = x^2$. Write this as the first term of the quotient.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ \end{array} $$

1. Multiply and subtract: Multiply the first term of the quotient ($x^2$) by the divisor ($x - 1$), which gives $x^2(x - 1) = x^3 - x^2$. Subtract this result from the dividend.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ \end{array} $$

1. Bring down the next term: Bring down the next term of the dividend ($+5x$).

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ \end{array} $$

1. Repeat the process: Now, divide the new leading term ($-2x^2$) by the leading term of the divisor ($x$). This gives $-2x^2 / x = -2x$. Write this as the next term of the quotient.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -2x \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ \end{array} $$

1. Multiply and subtract again: Multiply $-2x$ by the divisor ($x - 1$), which gives $-2x(x - 1) = -2x^2 + 2x$. Subtract this from the current polynomial.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -2x \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ -(-2x^2 & +2x) \\ \hline & & 3x & -3 \\ \end{array} $$

1. Bring down the last term: Bring down the last term of the dividend ($-3$).

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -2x \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ -(-2x^2 & +2x) \\ \hline & & 3x & -3 \\ \end{array} $$

1. Final division step: Divide the new leading term ($3x$) by the leading term of the divisor ($x$). This gives $3x / x = 3$. Write this as the last term of the quotient.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -2x & +3 \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ -(-2x^2 & +2x) \\ \hline & & 3x & -3 \\ \end{array} $$

1. Multiply and subtract one last time: Multiply $3$ by the divisor ($x - 1$), which gives $3(x - 1) = 3x - 3$. Subtract this from the current polynomial.

$$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -2x & +3 \\ x-1 \\ \hline x^3 & -3x^2 & +5x & -3 \\ -(x^3 & -x^2) \\ \hline & -2x^2 & +5x & \\ -(-2x^2 & +2x) \\ \hline & & 3x & -3 \\ -(3x & -3) \\ \hline & & & 0 \\ \end{array} $$

1. State the quotient: The remainder is 0, so the division is exact. The quotient is the polynomial written above the division line.