To find the other factor of the polynomial [tex]\( x^3 - 5x^2 \)[/tex] given that one of the factors is [tex]\( x + 3 \)[/tex], we proceed as follows:
1. Express the polynomial in factored form: We know that the given polynomial can be expressed as the product of two factors:
[tex]\[
x^3 - 5x^2 = (x + 3)(\text{other factor})
\][/tex]
2. Identify the other factor: Based on the problem, let's denote the other factor by [tex]\( g(x) \)[/tex]. Therefore, we must have:
[tex]\[
x^3 - 5x^2 = (x + 3)g(x)
\][/tex]
3. Determine the function [tex]\( g(x) \)[/tex]: Given a cubic polynomial and knowing one of its factors, we can find the other factor by factoring completely. The correct factorization of the polynomial [tex]\( x^3 - 5x^2 \)[/tex] is:
[tex]\[
x^3 - 5x^2 = x^2(x - 5)
\][/tex]
Therefore, the complete factorization shows that [tex]\( x^3 - 5x^2 \)[/tex] can be expressed as [tex]\( x^2(x - 5) \)[/tex]. This means the polynomial factors as [tex]\( x^2 \)[/tex] and [tex]\( (x - 5) \)[/tex].
4. Conclusion: Given one of the factors is [tex]\( x + 3 \)[/tex], it shows that the polynomial can also be written as a product involving [tex]\( x + 3 \)[/tex], which simplifies to include [tex]\( x - 5 \)[/tex] and [tex]\( x^2 \)[/tex].
Thus, if one of the factors of the polynomial [tex]\( x^3 - 5x^2 \)[/tex] is [tex]\( x + 3 \)[/tex], the other factor must be:
[tex]\[
x^2 (x - 5)
\][/tex]
However, considering the clarification, the primary correct other factor associated directly with [tex]\( x^3 - 5x^2 \)[/tex] is:
[tex]\[
x^2(x - 5)
\][/tex]
This results in [tex]\( x^2 \)[/tex] and [tex]\( (x - 5) \)[/tex] being correctly associated factors completing the polynomial factorization from where one of the stated factorizing [tex]\( x + 3 \)[/tex]:
Therefore,
[tex]\[
x^2(x - 5)
\][/tex] is the other term resulting in the given factorization yielding polynomial [tex]\( x^3 - 5x^2\)[/tex]
