Verified Answer :

  • Identify the greatest common factor (GCF) of \(-34 x^5 y^4 z^3\) and \(3 x z^4\), which is \(x z^3\).
  • Divide both terms by the GCF: \(x z^3 (-34 x^4 y^4 + 3 z)\).
  • Recognize that the expression within the parentheses cannot be factored further.
  • The final result is \(\boxed{x z^3 (3 z - 34 x^4 y^4)}\).

Explanation

  1. Identify the Greatest Common Factor (GCF)

To factor the expression \(-34 x^5 y^4 z^3 + 3 x z^4\), we first identify the greatest common factor (GCF) of the two terms.

  • The numerical coefficients are -34 and 3. The greatest common divisor of |-34| and 3 is 1.
  • For the variable \(x\), the first term contains \(x^5\) and the second term contains \(x^1\). The lowest power of \(x\) common to both is \(x^1\).
  • For the variable \(y\), it only appears in the first term, so it is not part of the GCF.
  • For the variable \(z\), the first term contains \(z^3\) and the second term contains \(z^4\). The lowest power of \(z\) common to both is \(z^3\).

Combining these, the GCF of the expression is \(x z^3\).

  1. Factor out the GCF

Now we factor out \(x z^3\) from each term in the expression:
\[-34 x^5 y^4 z^3 + 3 x z^4 = x z^3 \left( \frac{-34 x^5 y^4 z^3}{x z^3} + \frac{3 x z^4}{x z^3} \right)\]
Performing the divisions for each term:

  • \(\frac{-34 x^5 y^4 z^3}{x z^3} = -34 x^{5-1} y^4 z^{3-3} = -34 x^4 y^4\)
  • \(\frac{3 x z^4}{x z^3} = 3 x^{1-1} z^{4-3} = 3 z\)

Substituting these back into our factored expression gives:
\[x z^3 (-34 x^4 y^4 + 3 z)\]

  1. Final Verification and Conclusion

We check the remaining binomial \(-34 x^4 y^4 + 3 z\) to see if it can be factored further. There are no common factors between \(-34 x^4 y^4\) and \(3 z\), as their coefficients have no common divisor other than 1 and they share no common variables. Therefore, the expression is completely factored.
The final factored form is:
\[\boxed{x z^3 (3 z - 34 x^4 y^4)}\]

Examples

Factoring expressions is essential in fields like physics and engineering when simplifying complex formulas. For example, if you are modeling the trajectory of a projectile and encounter an expression like the one we just factored, identifying the common factors can significantly simplify the calculation of variables such as the object's speed at different times. By extracting shared quantities, you make it easier to solve for zero points or analyze the behavior of the system under different parameters.

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