To find the quotient of [tex]\((x^3 - 3x^2 + 5x - 3) \div (x - 1)\)[/tex], we need to perform polynomial long division. Here are the steps:
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
Write [tex]\(x^2\)[/tex] above the division symbol.
2. Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(x - 1\)[/tex]:
[tex]\[
x^2 \cdot (x - 1) = x^3 - x^2
\][/tex]
3. Subtract the result from the original polynomial:
[tex]\[
(x^3 - 3x^2 + 5x - 3) - (x^3 - x^2) = -3x^2 + x^2 + 5x - 3 = -2x^2 + 5x - 3
\][/tex]
4. Bring down the next term (which is 0 in this case) and repeat the process:
- Divide [tex]\(-2x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-2x^2}{x} = -2x
\][/tex]
Write [tex]\(-2x\)[/tex] above the division symbol next to [tex]\(x^2\)[/tex].
- Multiply [tex]\(-2x\)[/tex] by the divisor [tex]\(x - 1\)[/tex]:
[tex]\[
-2x \cdot (x - 1) = -2x^2 + 2x
\][/tex]
- Subtract this from the current remainder [tex]\(-2x^2 + 5x - 3\)[/tex]:
[tex]\[
(-2x^2 + 5x - 3) - (-2x^2 + 2x) = 5x - 2x - 3 = 3x - 3
\][/tex]
5. Repeat the process with [tex]\(3x - 3\)[/tex]:
- Divide [tex]\(3x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{3x}{x} = 3
\][/tex]
Write [tex]\(3\)[/tex] above the division symbol next to [tex]\(-2x\)[/tex].
- Multiply [tex]\(3\)[/tex] by the divisor [tex]\(x - 1\)[/tex]:
[tex]\[
3 \cdot (x - 1) = 3x - 3
\][/tex]
- Subtract this from the current remainder [tex]\(3x - 3\)[/tex]:
[tex]\[
(3x - 3) - (3x - 3) = 0
\][/tex]
Thus, the quotient is [tex]\(x^2 - 2x + 3\)[/tex] and the remainder is [tex]\(0\)[/tex].
So, the quotient of [tex]\((x^3 - 3x^2 + 5x - 3) \div (x - 1)\)[/tex] is:
[tex]\[ \boxed{x^2 - 2x + 3} \][/tex]
