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Chapter 5: Problem 47

Factor each trinomial. \(-15 a^{2}-70 a+120\)

Short Answer

Expert verified
\[ -15a^2 - 70a + 120 = -5(3a - 4)(a + 6) \]

Step by step solution

01

- Identify common factors

Look for the greatest common factor (GCF) of the coefficients of \text{-15, -70, and 120}. The GCF is 5, but since the leading coefficient is negative, factor out -5: \[ -15a^2 - 70a + 120 = -5(3a^2 + 14a - 24) \] This simplifies our trinomial.
02

- Factor the simplified trinomial

We now need to factor the trinomial within the parentheses: \(3a^2 + 14a - 24\). Look for two numbers that multiply to \(3 \times -24 = -72\) and add to 14. Those numbers are 18 and -4: \[ 3a^2 + 14a - 24 = 3a^2 + 18a - 4a - 24 \] Next, group the terms to factor by grouping: \[ (3a^2 + 18a) + (-4a - 24) \] Factor out the GCF from each group: \[ 3a(a + 6) - 4(a + 6) \] Notice that \(a + 6\) is common in both groups. Factor \(a + 6\) out: \[ (3a - 4)(a + 6) \]
03

- Write the final factored form

Combine the factored expression back with the common factor that was pulled out initially: \[ -5(3a^2 + 14a - 24) = -5(3a - 4)(a + 6) \] This is the fully factored form of the original trinomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
When factoring trinomials, the first step is often to identify and extract the Greatest Common Factor (GCF). The GCF is the largest number or variable that divides all the terms of the polynomial. This simplifies the trinomial and makes further factoring easier.
For example, consider the trinomial \(-15a^2 - 70a + 120\). The coefficients are -15, -70, and 120. The GCF of these numbers is 5. But since the leading coefficient is negative, factor out -5 instead to get: \[ -15a^2 - 70a + 120 = -5(3a^2 + 14a - 24) \]
  • Find the GCF of all terms.

  • Factor out the GCF.

  • If the leading coefficient is negative, factor out its negative GCF.
This transformation makes the polynomial simpler to handle in the following steps.
Factoring by Grouping
Factoring by grouping is a method used to factor polynomials with four terms by grouping pairs of terms together and factoring out their GCF. This technique is beneficial when simplifying the trinomial further.
Let's consider the simplified trinomial \3a^2 + 14a - 24. First, we find two numbers that multiply to \( 3 \times -24 = -72 \) and add to 14. Those numbers are 18 and -4: \[ 3a^2 + 14a - 24 = 3a^2 + 18a - 4a - 24 \]
Now, group the terms: \[ (3a^2 + 18a) + (-4a - 24) \]
Next, factor out the GCF from each group:
  • \

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