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Factor the following by grouping.

[tex]x^3 - 5x^2 + x - 5[/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]x^3 - 5x^2 + x - 5 = \boxed{}[/tex]

B. The polynomial is prime.


Sagot :

To factor the polynomial [tex]\( x^3 - 5x^2 + x - 5 \)[/tex], follow these steps:

1. Group the terms in pairs:
[tex]\[ (x^3 - 5x^2) + (x - 5) \][/tex]

2. Factor out the greatest common factor (GCF) from each group:

- In the first group, [tex]\( x^3 - 5x^2 \)[/tex], the GCF is [tex]\( x^2 \)[/tex]:
[tex]\[ x^2(x - 5) \][/tex]

- In the second group, [tex]\( x - 5 \)[/tex], there is no common factor other than 1:
[tex]\[ 1(x - 5) \][/tex]

3. Rewrite the polynomial with the factored terms:
[tex]\[ x^2(x - 5) + 1(x - 5) \][/tex]

4. Factor out the common binomial factor [tex]\((x - 5)\)[/tex]:
[tex]\[ (x - 5)(x^2 + 1) \][/tex]

Therefore, the polynomial [tex]\( x^3 - 5x^2 + x - 5 \)[/tex] factors as follows:
[tex]\[ \boxed{(x - 5)(x^2 + 1)} \][/tex]

So the correct choice is:
A. [tex]\( x^3 - 5x^2 + x - 5 = (x - 5)(x^2 + 1) \)[/tex]