Answer :
To factor the expression [tex]\(2x^3 - 5x^2 - 10x + 25\)[/tex] by grouping, let's follow these steps:
1. Group the terms:
Begin by grouping the terms in pairs:
[tex]\[(2x^3 - 5x^2) + (-10x + 25).\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group, [tex]\(2x^3 - 5x^2\)[/tex], the GCF is [tex]\(x^2\)[/tex]. Factor [tex]\(x^2\)[/tex] out:
[tex]\[x^2(2x - 5).\][/tex]
- In the second group, [tex]\(-10x + 25\)[/tex], the GCF is [tex]\(-5\)[/tex]. Factor [tex]\(-5\)[/tex] out:
[tex]\(-5(2x - 5).\]
3. Rewrite the expression:
Now, rewrite the expression using these factors:
\[x^2(2x - 5) - 5(2x - 5).\]
4. Factor out the common binomial:
Notice that both groups contain the common factor \(2x - 5\)[/tex]. Factor this out:
[tex]\[(x^2 - 5)(2x - 5).\][/tex]
So, the factored form of the expression [tex]\(2x^3 - 5x^2 - 10x + 25\)[/tex] is [tex]\((2x - 5)(x^2 - 5)\)[/tex].
1. Group the terms:
Begin by grouping the terms in pairs:
[tex]\[(2x^3 - 5x^2) + (-10x + 25).\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group, [tex]\(2x^3 - 5x^2\)[/tex], the GCF is [tex]\(x^2\)[/tex]. Factor [tex]\(x^2\)[/tex] out:
[tex]\[x^2(2x - 5).\][/tex]
- In the second group, [tex]\(-10x + 25\)[/tex], the GCF is [tex]\(-5\)[/tex]. Factor [tex]\(-5\)[/tex] out:
[tex]\(-5(2x - 5).\]
3. Rewrite the expression:
Now, rewrite the expression using these factors:
\[x^2(2x - 5) - 5(2x - 5).\]
4. Factor out the common binomial:
Notice that both groups contain the common factor \(2x - 5\)[/tex]. Factor this out:
[tex]\[(x^2 - 5)(2x - 5).\][/tex]
So, the factored form of the expression [tex]\(2x^3 - 5x^2 - 10x + 25\)[/tex] is [tex]\((2x - 5)(x^2 - 5)\)[/tex].
