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Answer :
To factor the trinomial [tex]\( 25x^3 - 55x^2 - 60x \)[/tex] by grouping, follow these steps:
1. Factor Out the Greatest Common Factor (GCF):
- First, identify the greatest common factor of all the terms. In this trinomial, the GCF is [tex]\( 5x \)[/tex].
- Factor out [tex]\( 5x \)[/tex] from each term:
[tex]\[
25x^3 - 55x^2 - 60x = 5x(5x^2 - 11x - 12)
\][/tex]
2. Focus on the Quadratic [tex]\( 5x^2 - 11x - 12 \)[/tex]:
- Now, we need to factor the quadratic expression inside the parentheses further by grouping.
3. Find Two Numbers That Multiply to [tex]\((5 \times -12)\)[/tex] and Add to [tex]\(-11\)[/tex]:
- We need numbers that multiply to [tex]\(-60\)[/tex] (the product of the coefficient of [tex]\( x^2 \)[/tex] which is 5, and the constant term [tex]\(-12\)[/tex]) and add to [tex]\(-11\)[/tex].
- These numbers are [tex]\(-15\)[/tex] and [tex]\(4\)[/tex].
4. Rewrite the Middle Term Using These Numbers:
- Rewrite [tex]\(-11x\)[/tex] as [tex]\(-15x + 4x\)[/tex]:
[tex]\[
5x^2 - 11x - 12 = 5x^2 - 15x + 4x - 12
\][/tex]
5. Group the Terms and Factor Them Separately:
- Group the terms into two pairs: [tex]\((5x^2 - 15x)\)[/tex] and [tex]\((4x - 12)\)[/tex].
- Factor each pair:
[tex]\[
5x(x - 3) + 4(x - 3)
\][/tex]
6. Factor by Grouping:
- Notice that each group contains a common binomial factor [tex]\((x - 3)\)[/tex].
- Factor out the common binomial:
[tex]\[
(5x + 4)(x - 3)
\][/tex]
7. Final Factored Form:
- Combine the factored expression with the GCF from step 1:
[tex]\[
5x(5x + 4)(x - 3)
\][/tex]
Thus, the trinomial [tex]\( 25x^3 - 55x^2 - 60x \)[/tex] is factored as [tex]\( 5x(x - 3)(5x + 4) \)[/tex].
1. Factor Out the Greatest Common Factor (GCF):
- First, identify the greatest common factor of all the terms. In this trinomial, the GCF is [tex]\( 5x \)[/tex].
- Factor out [tex]\( 5x \)[/tex] from each term:
[tex]\[
25x^3 - 55x^2 - 60x = 5x(5x^2 - 11x - 12)
\][/tex]
2. Focus on the Quadratic [tex]\( 5x^2 - 11x - 12 \)[/tex]:
- Now, we need to factor the quadratic expression inside the parentheses further by grouping.
3. Find Two Numbers That Multiply to [tex]\((5 \times -12)\)[/tex] and Add to [tex]\(-11\)[/tex]:
- We need numbers that multiply to [tex]\(-60\)[/tex] (the product of the coefficient of [tex]\( x^2 \)[/tex] which is 5, and the constant term [tex]\(-12\)[/tex]) and add to [tex]\(-11\)[/tex].
- These numbers are [tex]\(-15\)[/tex] and [tex]\(4\)[/tex].
4. Rewrite the Middle Term Using These Numbers:
- Rewrite [tex]\(-11x\)[/tex] as [tex]\(-15x + 4x\)[/tex]:
[tex]\[
5x^2 - 11x - 12 = 5x^2 - 15x + 4x - 12
\][/tex]
5. Group the Terms and Factor Them Separately:
- Group the terms into two pairs: [tex]\((5x^2 - 15x)\)[/tex] and [tex]\((4x - 12)\)[/tex].
- Factor each pair:
[tex]\[
5x(x - 3) + 4(x - 3)
\][/tex]
6. Factor by Grouping:
- Notice that each group contains a common binomial factor [tex]\((x - 3)\)[/tex].
- Factor out the common binomial:
[tex]\[
(5x + 4)(x - 3)
\][/tex]
7. Final Factored Form:
- Combine the factored expression with the GCF from step 1:
[tex]\[
5x(5x + 4)(x - 3)
\][/tex]
Thus, the trinomial [tex]\( 25x^3 - 55x^2 - 60x \)[/tex] is factored as [tex]\( 5x(x - 3)(5x + 4) \)[/tex].
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