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Answer :
To factor the expression [tex]\(x^3 - 5x^2 + 5x - 25\)[/tex] by grouping, follow these steps:
1. Group the terms: Start by grouping the terms in pairs:
[tex]\[
(x^3 - 5x^2) + (5x - 25)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\((x^3 - 5x^2)\)[/tex], factor out [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x - 5)
\][/tex]
- From the second group [tex]\((5x - 25)\)[/tex], factor out 5:
[tex]\[
5(x - 5)
\][/tex]
3. Notice the common factor: Both groups now have a common factor of [tex]\((x - 5)\)[/tex].
4. Factor out the common binomial:
- Combine the [tex]\(x^2\)[/tex] and [tex]\(5\)[/tex] into one expression, using the common factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x^2 + 5)(x - 5)
\][/tex]
Thus, the expression [tex]\(x^3 - 5x^2 + 5x - 25\)[/tex] factors to [tex]\((x - 5)(x^2 + 5)\)[/tex]. This is the fully factored form using grouping.
1. Group the terms: Start by grouping the terms in pairs:
[tex]\[
(x^3 - 5x^2) + (5x - 25)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\((x^3 - 5x^2)\)[/tex], factor out [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x - 5)
\][/tex]
- From the second group [tex]\((5x - 25)\)[/tex], factor out 5:
[tex]\[
5(x - 5)
\][/tex]
3. Notice the common factor: Both groups now have a common factor of [tex]\((x - 5)\)[/tex].
4. Factor out the common binomial:
- Combine the [tex]\(x^2\)[/tex] and [tex]\(5\)[/tex] into one expression, using the common factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x^2 + 5)(x - 5)
\][/tex]
Thus, the expression [tex]\(x^3 - 5x^2 + 5x - 25\)[/tex] factors to [tex]\((x - 5)(x^2 + 5)\)[/tex]. This is the fully factored form using grouping.
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