Question

question
factor completely:

( 15x^2 - 12x - 3x^3 )

answer
attempt 1 out of 2

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Explanation:

Step1: Rearrange the polynomial

First, we rearrange the given polynomial \(15x^{2}-12x - 3x^{3}\) in standard form (descending powers of \(x\)): \(-3x^{3}+15x^{2}-12x\). We can factor out a negative sign to make the leading coefficient positive: \(-(3x^{3}-15x^{2}+12x)\).

Step2: Factor out the GCF

Next, we find the greatest common factor (GCF) of the terms inside the parentheses. The GCF of \(3x^{3}\), \(15x^{2}\), and \(12x\) is \(3x\). Factoring out \(3x\) from \(3x^{3}-15x^{2}+12x\), we get:
\(3x(x^{2}-5x + 4)\)

Step3: Factor the quadratic

Now, we factor the quadratic \(x^{2}-5x + 4\). We need two numbers that multiply to \(4\) and add up to \(-5\). The numbers are \(-1\) and \(-4\). So, \(x^{2}-5x + 4=(x - 1)(x - 4)\).

Step4: Combine all factors

Putting it all together, we have:
\(-(3x(x - 1)(x - 4))=-3x(x - 1)(x - 4)\) (or we can also write it as \(3x(-(x - 1))(x - 4)=3x(1 - x)(x - 4)\), but the first form is also correct)

Answer:

\(-3x(x - 1)(x - 4)\) (or \(3x(1 - x)(x - 4)\) or \(3x(x - 4)(1 - x)\) or other equivalent factored forms)