To divide [tex]\( \left(4 x^3 - 13 x^2 - 5 x\right) \)[/tex] by [tex]\( \left(x^2 - 3 x - 2\right) \)[/tex] using polynomial long division, follow these detailed steps:
1. Setup: Write the division in the long division format with [tex]\( 4 x^3 - 13 x^2 - 5 x \)[/tex] inside the division symbol and [tex]\( x^2 - 3 x - 2 \)[/tex] outside.
2. First Term: Divide the first term of the numerator [tex]\( 4 x^3 \)[/tex] by the first term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{4 x^3}{x^2} = 4 x
\][/tex]
3. Multiply and Subtract: Multiply [tex]\( 4 x \)[/tex] by the entire divisor [tex]\( x^2 - 3 x - 2 \)[/tex] and subtract:
[tex]\[
4 x \cdot (x^2 - 3 x - 2) = 4 x^3 - 12 x^2 - 8 x
\][/tex]
Subtraction:
[tex]\[
(4 x^3 - 13 x^2 - 5 x) - (4 x^3 - 12 x^2 - 8 x) = -x^2 + 3 x
\][/tex]
4. Second Term: Now take the new polynomial [tex]\( -x^2 + 3 x \)[/tex] and repeat the process: Divide [tex]\( -x^2 \)[/tex] by [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{-x^2}{x^2} = -1
\][/tex]
5. Multiply and Subtract: Multiply [tex]\( -1 \)[/tex] by the entire divisor [tex]\( x^2 - 3 x - 2 \)[/tex] and subtract:
[tex]\[
-1 \cdot (x^2 - 3 x - 2) = -x^2 + 3 x + 2
\][/tex]
Subtraction:
[tex]\[
(-x^2 + 3 x) - (-x^2 + 3 x + 2) = -2
\][/tex]
6. Result: The quotient is the combination of our results from each step:
[tex]\[
4 x - 1
\][/tex]
And the remainder is:
[tex]\[
-2
\][/tex]
Therefore, the result of dividing [tex]\( \left(4 x^3 - 13 x^2 - 5 x\right) \)[/tex] by [tex]\( \left(x^2 - 3 x - 2\right) \)[/tex] is:
[tex]\[
\boxed{4 x - 1 \text{ with a remainder of } -2}
\][/tex]
