To solve the polynomial division [tex]\(\left(3x^3 - 4x^2 + 3x - 2\right) \div (x - 2)\)[/tex], we proceed as follows:
1. Set up the Polynomial Division:
[tex]\[
\frac{3x^3 - 4x^2 + 3x - 2}{x - 2}
\][/tex]
2. Divide the Leading Terms:
The first term in the quotient is obtained by dividing the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^3}{x} = 3x^2
\][/tex]
So [tex]\(\mathbf{Q(x) = 3x^2}\)[/tex].
3. Multiply and Subtract:
Multiply [tex]\(3x^2\)[/tex] by [tex]\(x - 2\)[/tex] and subtract the result from the original polynomial:
[tex]\[
(3x^2)(x - 2) = 3x^3 - 6x^2
\][/tex]
[tex]\[
3x^3 - 4x^2 + 3x - 2 - (3x^3 - 6x^2) = 2x^2 + 3x - 2
\][/tex]
4. Repeat the Process:
Next, divide [tex]\(2x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
[tex]\(\mathbf{Q(x) = 3x^2 - 2x}\)[/tex].
Multiply [tex]\(2x\)[/tex] by [tex]\(x - 2\)[/tex] and subtract the result:
[tex]\[
(2x)(x - 2) = 2x^2 - 4x
\][/tex]
[tex]\[
2x^2 + 3x - 2 - (2x^2 - 4x) = 7x - 2
\][/tex]
5. Continue the Process:
Divide [tex]\(7x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{7x}{x} = 7
\][/tex]
So [tex]\(\mathbf{Q(x) = 3x^2 - 2x + 7}\)[/tex].
Multiply [tex]\(7\)[/tex] by [tex]\(x - 2\)[/tex] and subtract:
[tex]\[
7(x - 2) = 7x - 14
\][/tex]
[tex]\[
7x - 2 - (7x - 14) = 12
\][/tex]
6. Remainder:
The remainder which cannot be divided by [tex]\(x - 2\)[/tex] is [tex]\(12\)[/tex].
Hence, the final quotient and remainder are:
[tex]\[
Q(x) = 3x^2 - 2x + 7
\][/tex]
[tex]\[
R(x) = 12
\][/tex]
Among the provided options, the correct one matches:
[tex]\[
Q(x) = 3x^2 + 2x + 7 ; R(x) = 12
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
