- Identify the GCF of the expression \(6n^4 - 24n^3 + 18n\), which is \(6n\).
- Factor out \(6n\) to rewrite the expression as \(6n(n^3 - 4n^2 + 3)\).
- Compare this to the options and identify the correct choice.
- The correct factored form is \(\boxed{6n(n^3 - 4n^2 + 3)}\).
Explanation
- Analysis of the expression and identifying the GCF
The expression to be factored is \(6n^4 - 24n^3 + 18n\). Our objective is to determine which of the provided options is equivalent to this expression by factoring out the greatest common factor (GCF). The terms are \(6n^4\), \(-24n^3\), and \(18n\). To find the GCF, we look at the coefficients (6, -24, 18) and the powers of \(n\) (\(n^4, n^3, n^1\)). The GCF of the coefficients is 6, and the GCF of the variables is \(n^1\). Thus, the GCF is \(6n\).
- Factoring the expression
We factor out \(6n\) from the original expression \(6n^4 - 24n^3 + 18n\):
\[6n^4 - 24n^3 + 18n = 6n \cdot (n^3 - 4n^2 + 3)\]
By dividing each term by \(6n\), we get:
- \(\frac{6n^4}{6n} = n^3\)
- \(\frac{-24n^3}{6n} = -4n^2\)
- \(\frac{18n}{6n} = 3\)
This confirms that the factored form is \(6n(n^3 - 4n^2 + 3)\).
- Comparing with given options
Comparing our result \(6n(n^3 - 4n^2 + 3)\) with the provided options:
- \(6n(n^4 + 4n^3 + 3n)\)
- \(6n(n^4 - 4n^3 + 3n)\)
- \(6n(n^3 - 4n^2 + 3)\)
- \(6n(n^3 + 4n^2 + 3)\)
The third option matches our result. Thus, the correct factored form is \(6n(n^3 - 4n^2 + 3)\).
Examples
Factoring expressions is essential in fields like engineering and physics, where we often need to simplify complex equations to find roots or analyze behaviors. For example, if you are modeling the trajectory of a projectile where the height \(h\) at time \(t\) is given by a polynomial, factoring can help you determine the exact times \(t\) when the object hits the ground (i.e., when \(h = 0\)). Just as we factored \(6n^4 - 24n^3 + 18n\) to find a simpler product form, you can simplify and solve real-world physics problems more efficiently.