To determine the other factor of the polynomial [tex]\( x^3 - 5x^2 \)[/tex] given that one of the factors is [tex]\( x + 3 \)[/tex], we need to perform polynomial division.
Here's how the steps of the polynomial division process work:
1. Divide the highest degree term of the polynomial by the highest degree term of the divisor:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
2. Multiply the result by the divisor and subtract it from the original polynomial to find the new polynomial to be divided:
[tex]\[
(x^2) \cdot (x + 3) = x^3 + 3x^2
\][/tex]
[tex]\[
x^3 - 5x^2 - (x^3 + 3x^2) = -8x^2
\][/tex]
3. Repeat the division with the new polynomial:
[tex]\[
\frac{-8x^2}{x} = -8x
\][/tex]
4. Multiply the result by the divisor and subtract it from the polynomial:
[tex]\[
(-8x) \cdot (x + 3) = -8x^2 - 24x
\][/tex]
[tex]\[
-8x^2 - (-8x^2 - 24x) = 24x
\][/tex]
5. Repeat the division with the new polynomial:
[tex]\[
\frac{24x}{x} = 24
\][/tex]
6. Multiply the result by the divisor and subtract it from the polynomial:
[tex]\[
(24) \cdot (x + 3) = 24x + 72
\][/tex]
[tex]\[
24x - (24x + 72) = -72
\][/tex]
7. The result of the division gives us the quotient [tex]\( x^2 - 8x + 24 \)[/tex] and a remainder of [tex]\( -72 \)[/tex].
So the polynomial [tex]\( x^3 - 5x^2 \)[/tex] can be expressed as:
[tex]\[
(x + 3)(x^2 - 8x + 24) + (-72)
\][/tex]
Thus, the closest option without the remainder factor is:
[tex]\[
\boxed{x^2 - 8x + 24}
\][/tex]
Therefore, the correct answer is:
B. [tex]\( x^2 - 8x + 24 \)[/tex]
