- Extract the common factor \(2x\) from the polynomial, resulting in \(2x(x^4 + 6x^2 - 27)\).
- Substitute \(u = x^2\) to treat the quartic expression as a quadratic \(u^2 + 6u - 27\).
- Factor the quadratic as \((u + 9)(u - 3)\) using the sum and product method.
- Return to the original variable to express the final factored form as \(\boxed{2x(x^2 - 3)(x^2 + 9)}\).
Explanation
- Factor out the common term
To find the completely factored form of the polynomial \(P(x) = 2x^5 + 12x^3 - 54x\), we first identify the greatest common factor among all terms. Notice that each term is divisible by \(2x\). We factor out \(2x\):
\[2x^5 + 12x^3 - 54x = 2x(x^4 + 6x^2 - 27).\]
- Factor the quadratic expression using substitution
Now we focus on factoring the quartic expression inside the parentheses: \(x^4 + 6x^2 - 27\). This expression is a quadratic in terms of \(x^2\). Let \(u = x^2\). Then the expression becomes:
\[u^2 + 6u - 27.\]
We need to find two numbers whose product is \(-27\) and whose sum is \(6\). Testing the factors of \(-27\), we find that \(9\) and \(-3\) satisfy these conditions, since \(9 \times (-3) = -27\) and \(9 + (-3) = 6\). Therefore, we can write:
\[u^2 + 6u - 27 = (u + 9)(u - 3).\]
- Substitute back and conclude the factorization
Now, substitute \(x^2\) back for \(u\) in the expression we just factored:
\[(u + 9)(u - 3) = (x^2 + 9)(x^2 - 3).\]
Combining this with the common factor \(2x\) we pulled out in the first step, the complete factorization is:
\[2x(x^2 - 3)(x^2 + 9).\]
Comparing this result with the given options, it matches the fourth option: \(2x(x^2 - 3)(x^2 + 9)\).
Examples
Factoring polynomials is a fundamental skill used in many areas of engineering and physics. For instance, in structural engineering, factoring polynomials helps determine the critical points of a load distribution curve on a bridge beam, allowing engineers to identify where stress is highest or zero. This ensures the design is both safe and material-efficient, demonstrating how abstract algebraic techniques are applied to build stable and functional real-world structures.