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Chapter 5: Problem 69

Factor completely. $$ -6 x^{4}+8 x^{3}-12 x $$

Short Answer

Expert verified
\(-2x(-3x^3 + 4x^2 - 6)\)

Step by step solution

01

Identify common factors

Look for the greatest common factor (GCF) among the terms \(-6x^4, 8x^3, -12x\). In this case, the GCF is \(-2x\).
02

Factor out the GCF

Factor \(-2x\) out of each term. This gives us: \-2x(-3x^3 + 4x^2 - 6)\.
03

Factor the remaining polynomial

We need to factor the polynomial \(-3x^3 + 4x^2 - 6\) completely. Start by observing if simple factoring methods like grouping apply. In this context, it turns out that the remaining polynomial cannot be factored further using elementary methods.

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Key Concepts

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

greatest common factor (GCF)
The Greatest Common Factor (GCF) is the highest factor that divides all the terms in a polynomial without leaving a remainder. Finding the GCF is the first step in factoring polynomials. In the given exercise, the terms are \( -6x^4, 8x^3, -12x \). To determine the GCF, follow these steps:
  • Identify the numerical factors for each term: \(-6, 8, -12\)
  • Find the common factors among these numeric values: Both 6 and 12 are divisible by 2, while 8 is divisible by 2. The highest such common factor is 2.
  • Identify the smallest power of the variable present in all terms: Here, the variable is \(x\), and the lowest power is 1 (\

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