Answered

What is the completely factored form of $3 x^5-7 x^4+6 x^2-14 x$ ?


A$\left(x^4+2 x\right)(3 x-7)$
B$x^4(3 x-7)(2 x-1)$
C$x\left(x^3+2\right)(3 x-7)$
D$x\left(3 x^4-7 x^3+6 x-14\right)$

Verified Answer :

  • Start by factoring out the common monomial \(x\) from \(3x^5 - 7x^4 + 6x^2 - 14x\) to get \(x(3x^4 - 7x^3 + 6x - 14)\).
  • Group the terms within the parentheses as \((3x^4 - 7x^3) + (6x - 14)\).
  • Factor out \(x^3\) from the first group and \(2\) from the second group to obtain \(x^3(3x - 7) + 2(3x - 7)\).
  • Extract the common binomial \((3x - 7)\) to reach the final form \(\boxed{x(x^3 + 2)(3x - 7)}\).

Explanation

  1. Factoring out the Common Monomial Factor

We are asked to find the completely factored form of the polynomial \(P(x) = 3x^5 - 7x^4 + 6x^2 - 14x\). To factor this expression, we first look for common factors among all terms. Each term contains at least one power of \(x\), so we can factor out \(x\).
\[3x^5 - 7x^4 + 6x^2 - 14x = x(3x^4 - 7x^3 + 6x - 14)\]
Now our goal is to factor the quartic polynomial inside the parentheses: \(3x^4 - 7x^3 + 6x - 14\).

  1. Factoring by Grouping

To factor the quartic polynomial \(3x^4 - 7x^3 + 6x - 14\), we try factoring by grouping. We can group the terms as follows:
\[(3x^4 - 7x^3) + (6x - 14)\]
Now, we look for common factors in each group.

  • In the first group, \(3x^4 - 7x^3\), the greatest common factor is \(x^3\), so we get \(x^3(3x - 7)\).
  • In the second group, \(6x - 14\), the greatest common factor is \(2\), so we get \(2(3x - 7)\).

Combining these, we have:
\[x^3(3x - 7) + 2(3x - 7)\]
Notice that both terms now share the common binomial factor \((3x - 7)\).

  1. Final Factorization and Comparison

We can now factor out the common binomial \((3x - 7)\) from the expression \(x^3(3x - 7) + 2(3x - 7)\):
\[(x^3 + 2)(3x - 7)\]
Combining this with the common factor \(x\) we pulled out in the first step, the complete factorization is:
\[x(x^3 + 2)(3x - 7)\]
Comparing this result to the given options:

  1. \((x^4 + 2x)(3x - 7)\)
  2. \(x^4(3x - 7)(2x - 1)\)
  3. \(x(x^3 + 2)(3x - 7)\)
  4. \(x(3x^4 - 7x^3 + 6x - 14)\)

Our result matches the third option.

Examples

Factoring is a fundamental skill in algebra that allows us to simplify complex expressions and solve equations more efficiently. For instance, in engineering, when analyzing the motion of an object described by a high-degree polynomial, factoring can help identify critical points where the velocity or acceleration might be zero, making it much easier to predict the system's behavior. Mastering these techniques is essential for solving higher-order equations in physics, economics, and data analysis.

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