To perform the division [tex]\( \left(3x^3 - 5x^2 - 17x - 5\right) \div (3x + 1) \)[/tex], we will apply polynomial long division. Here are the step-by-step details of the process:
1. Set up the division:
We write [tex]\( 3x^3 - 5x^2 - 17x - 5 \)[/tex] as the dividend and [tex]\( 3x + 1 \)[/tex] as the divisor.
2. First step:
Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{3x^3}{3x} = x^2
\][/tex]
Multiply the entire divisor by [tex]\( x^2 \)[/tex] and subtract it from the dividend:
[tex]\[
(3x^3 - 5x^2 - 17x - 5) - (3x^3 + x^2) = -6x^2 - 17x - 5
\][/tex]
3. Second step:
Divide the new leading term of the current dividend by the first term of the divisor:
[tex]\[
\frac{-6x^2}{3x} = -2x
\][/tex]
Multiply the entire divisor by [tex]\( -2x \)[/tex] and subtract it from the current dividend:
[tex]\[
(-6x^2 - 17x - 5) - (-6x^2 - 2x) = -15x - 5
\][/tex]
4. Third step:
Divide the new leading term of the current dividend by the first term of the divisor:
[tex]\[
\frac{-15x}{3x} = -5
\][/tex]
Multiply the entire divisor by [tex]\( -5 \)[/tex] and subtract it from the current dividend:
[tex]\[
(-15x - 5) - (-15x - 5) = 0
\][/tex]
Thus, we find that the quotient is [tex]\( x^2 - 2x - 5 \)[/tex] and the remainder is 0.
Therefore, the final answer is:
[tex]\[
x^2 - 2x - 5 + \frac{0}{3x + 1}
\][/tex]
Since the remainder is zero, we can simplify this to:
[tex]\[
x^2 - 2x - 5
\][/tex]
