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Answer :
We want to factor the expression
[tex]$$
3x^3 - 5x^2 - 15x + 25
$$[/tex]
by grouping. Here is the step-by-step solution:
1. Group the terms into two pairs:
[tex]$$
(3x^3 - 5x^2) + (-15x + 25).
$$[/tex]
2. Factor out the greatest common factor from each group:
• From the first group, [tex]\(3x^3 - 5x^2\)[/tex], factor out [tex]\(x^2\)[/tex]:
[tex]$$
3x^3 - 5x^2 = x^2(3x - 5).
$$[/tex]
• From the second group, [tex]\(-15x + 25\)[/tex], factor out [tex]\(-5\)[/tex]:
[tex]$$
-15x + 25 = -5(3x - 5).
$$[/tex]
3. Notice that both groups contain the common factor [tex]\((3x - 5)\)[/tex]. Factor this common factor out:
[tex]$$
3x^3 - 5x^2 - 15x + 25 = (3x - 5)(x^2 - 5).
$$[/tex]
Thus, the expression factors as
[tex]$$
\boxed{(3x - 5)(x^2 - 5)}.
$$[/tex]
This is the final, factored form of the polynomial.
[tex]$$
3x^3 - 5x^2 - 15x + 25
$$[/tex]
by grouping. Here is the step-by-step solution:
1. Group the terms into two pairs:
[tex]$$
(3x^3 - 5x^2) + (-15x + 25).
$$[/tex]
2. Factor out the greatest common factor from each group:
• From the first group, [tex]\(3x^3 - 5x^2\)[/tex], factor out [tex]\(x^2\)[/tex]:
[tex]$$
3x^3 - 5x^2 = x^2(3x - 5).
$$[/tex]
• From the second group, [tex]\(-15x + 25\)[/tex], factor out [tex]\(-5\)[/tex]:
[tex]$$
-15x + 25 = -5(3x - 5).
$$[/tex]
3. Notice that both groups contain the common factor [tex]\((3x - 5)\)[/tex]. Factor this common factor out:
[tex]$$
3x^3 - 5x^2 - 15x + 25 = (3x - 5)(x^2 - 5).
$$[/tex]
Thus, the expression factors as
[tex]$$
\boxed{(3x - 5)(x^2 - 5)}.
$$[/tex]
This is the final, factored form of the polynomial.
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