- Identify the Greatest Common Factor of \(18x^3, -120x^2, -42x\), which is \(6x\).
- Rewrite the polynomial as \(6x(3x^2 - 20x - 7)\).
- Factor the quadratic part \(3x^2 - 20x - 7\) by splitting the middle term into \(-21x + x\), yielding \((3x+1)(x-7)\).
- Combine the GCF with the linear factors to reach the final expression: \(\boxed{6x(3x+1)(x-7)}\).
Explanation
- Identifying the Greatest Common Factor
The given polynomial is \(18x^3 - 120x^2 - 42x\). To factor it completely, we first look for the Greatest Common Factor (GCF) of all its terms. The coefficients are \(18\), \(-120\), and \(-42\). We can identify the GCF by looking at the prime factorization of these coefficients:
\(18 = 2 \times 3 \times 3\)
\(120 = 2 \times 2 \times 2 \times 3 \times 5\)
\(42 = 2 \times 3 \times 7\)
The common factors are \(2\) and \(3\), so the numerical GCF is \(2 \times 3 = 6\). Since every term also contains at least one power of \(x\), the total GCF of the polynomial is \(6x\).
- Factoring the Quadratic Expression
We now factor out the GCF \(6x\) from the original polynomial:
\(18x^3 - 120x^2 - 42x = 6x(3x^2 - 20x - 7)\)
Our next task is to factor the quadratic trinomial \(3x^2 - 20x - 7\) into two linear factors. We look for two integers that multiply to \(3 \times (-7) = -21\) and add to \(-20\). By observing the factors of \(-21\), we find that \(-21\) and \(1\) satisfy these conditions since \((-21) \times 1 = -21\) and \((-21) + 1 = -20\).
- Completing the Factorization
Using the integers \(-21\) and \(1\), we split the middle term of the quadratic trinomial:
\(3x^2 - 20x - 7 = 3x^2 - 21x + x - 7\)
Next, we factor by grouping:
\(3x^2 - 21x + x - 7 = 3x(x - 7) + 1(x - 7)\)
Factoring out the common binomial \((x - 7)\), we obtain:
\((3x + 1)(x - 7)\)
Combining this with our GCF, the completely factored form of the original polynomial is \(6x(3x + 1)(x - 7)\).
Examples
Factoring polynomials is a fundamental skill in algebra with applications in physics and engineering. For example, if the path of a projectile is described by a cubic function, factoring the expression allows engineers to easily determine the specific times when the projectile hits the ground (the roots). By breaking a complex expression down into simpler linear parts, we gain a clearer understanding of the underlying dynamics of the system.