Answer :
To find the quotient of the polynomial division [tex]\((x^3 - 3x^2 + 5x - 3) \div (x - 1)\)[/tex]:
1. Setup the division problem: We begin with the dividend [tex]\(x^3 - 3x^2 + 5x - 3\)[/tex] and the divisor [tex]\(x - 1\)[/tex].
2. Divide the leading terms: The leading term of the dividend is [tex]\(x^3\)[/tex] and the leading term of the divisor is [tex]\(x\)[/tex]. Divide [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex] to get the first term of the quotient, which is [tex]\(x^2\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x^2\)[/tex] by the divisor [tex]\(x - 1\)[/tex] giving [tex]\(x^3 - x^2\)[/tex]. Subtract [tex]\(x^3 - x^2\)[/tex] from the original dividend [tex]\(x^3 - 3x^2 + 5x - 3\)[/tex]:
[tex]\[ (x^3 - 3x^2 + 5x - 3) - (x^3 - x^2) = -2x^2 + 5x - 3 \][/tex]
4. Repeat the process: Divide the leading term [tex]\(-2x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-2x\)[/tex]. Multiply [tex]\(-2x\)[/tex] by the divisor [tex]\(x - 1\)[/tex] giving [tex]\(-2x^2 + 2x\)[/tex]. Subtract [tex]\(-2x^2 + 2x\)[/tex] from [tex]\(-2x^2 + 5x - 3\)[/tex]:
[tex]\[ (-2x^2 + 5x - 3) - (-2x^2 + 2x) = 3x - 3 \][/tex]
5. Final division: Divide the leading term [tex]\(3x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3\)[/tex]. Multiply [tex]\(3\)[/tex] by the divisor [tex]\(x - 1\)[/tex] giving [tex]\(3x - 3\)[/tex]. Subtract [tex]\(3x - 3\)[/tex] from [tex]\(3x - 3\)[/tex]:
[tex]\[ (3x - 3) - (3x - 3) = 0 \][/tex]
Since there is no remainder, the quotient of the division is [tex]\(x^2 - 2x + 3\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{x^2 - 2x + 3} \][/tex]
1. Setup the division problem: We begin with the dividend [tex]\(x^3 - 3x^2 + 5x - 3\)[/tex] and the divisor [tex]\(x - 1\)[/tex].
2. Divide the leading terms: The leading term of the dividend is [tex]\(x^3\)[/tex] and the leading term of the divisor is [tex]\(x\)[/tex]. Divide [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex] to get the first term of the quotient, which is [tex]\(x^2\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x^2\)[/tex] by the divisor [tex]\(x - 1\)[/tex] giving [tex]\(x^3 - x^2\)[/tex]. Subtract [tex]\(x^3 - x^2\)[/tex] from the original dividend [tex]\(x^3 - 3x^2 + 5x - 3\)[/tex]:
[tex]\[ (x^3 - 3x^2 + 5x - 3) - (x^3 - x^2) = -2x^2 + 5x - 3 \][/tex]
4. Repeat the process: Divide the leading term [tex]\(-2x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-2x\)[/tex]. Multiply [tex]\(-2x\)[/tex] by the divisor [tex]\(x - 1\)[/tex] giving [tex]\(-2x^2 + 2x\)[/tex]. Subtract [tex]\(-2x^2 + 2x\)[/tex] from [tex]\(-2x^2 + 5x - 3\)[/tex]:
[tex]\[ (-2x^2 + 5x - 3) - (-2x^2 + 2x) = 3x - 3 \][/tex]
5. Final division: Divide the leading term [tex]\(3x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3\)[/tex]. Multiply [tex]\(3\)[/tex] by the divisor [tex]\(x - 1\)[/tex] giving [tex]\(3x - 3\)[/tex]. Subtract [tex]\(3x - 3\)[/tex] from [tex]\(3x - 3\)[/tex]:
[tex]\[ (3x - 3) - (3x - 3) = 0 \][/tex]
Since there is no remainder, the quotient of the division is [tex]\(x^2 - 2x + 3\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{x^2 - 2x + 3} \][/tex]
