To find the quotient of the polynomial division [tex]\(\left(x^3 - 3x^2 + 5x - 3\right) \div (x - 1)\)[/tex], we will perform polynomial long division.
### Step-by-Step Solution:
1. Set Up the Division:
We divide the polynomial [tex]\(x^3 - 3x^2 + 5x - 3\)[/tex] by [tex]\(x - 1\)[/tex].
2. Divide the First Term:
Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
So, the first term of the quotient is [tex]\(x^2\)[/tex].
3. Multiply and Subtract:
Multiply [tex]\(x^2\)[/tex] by the divisor [tex]\(x - 1\)[/tex] and subtract from the dividend:
[tex]\[
(x^3 - 3x^2 + 5x - 3) - (x^2 \cdot (x - 1)) = (x^3 - 3x^2 + 5x - 3) - (x^3 - x^2) = -2x^2 + 5x - 3
\][/tex]
4. Repeat the Process:
Continue with [tex]\(-2x^2 + 5x - 3\)[/tex]:
[tex]\[
\frac{-2x^2}{x} = -2x
\][/tex]
So, the next term in the quotient is [tex]\(-2x\)[/tex].
Multiply [tex]\(-2x\)[/tex] by [tex]\(x - 1\)[/tex] and subtract:
[tex]\[
(-2x^2 + 5x - 3) - (-2x \cdot (x - 1)) = (-2x^2 + 5x - 3) - (-2x^2 + 2x) = 3x - 3
\][/tex]
5. Continue:
Continue with [tex]\(3x - 3\)[/tex]:
[tex]\[
\frac{3x}{x} = 3
\][/tex]
So, the next term in the quotient is [tex]\(3\)[/tex].
Multiply [tex]\(3\)[/tex] by [tex]\(x - 1\)[/tex] and subtract:
[tex]\[
(3x - 3) - (3 \cdot (x - 1)) = (3x - 3) - (3x - 3) = 0
\][/tex]
6. Conclusion:
The quotient of [tex]\(\left(x^3 - 3x^2 + 5x - 3\right) \div (x - 1)\)[/tex] is:
[tex]\[
x^2 - 2x + 3
\][/tex]
So, the correct answer is:
[tex]\[
x^2 - 2x + 3
\][/tex]
