Question

question
factor completely:
11x² - 24x - x³
answer attempt 1 out of 2
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Explanation:

Step1: Rearrange the polynomial

First, we rewrite the polynomial in standard form (descending powers of \(x\)): \(-x^{3}+11x^{2}-24x\). We can factor out a negative sign to make the leading coefficient positive: \(-(x^{3}-11x^{2}+24x)\).

Step2: Factor out the GCF

Next, we factor out the greatest common factor (GCF) from the polynomial inside the parentheses. The GCF of \(x^{3}\), \(-11x^{2}\), and \(24x\) is \(x\). So we have: \(-x(x^{2}-11x + 24)\).

Step3: Factor the quadratic

Now we need to factor the quadratic \(x^{2}-11x + 24\). We look for two numbers that multiply to \(24\) and add up to \(-11\) (but since the middle term is \(-11x\) and the constant term is positive, both numbers are negative). The numbers \(-3\) and \(-8\) work because \((-3)\times(-8)=24\) and \((-3)+(-8)=-11\). So we can factor the quadratic as \((x - 3)(x - 8)\).

Step4: Combine all factors

Putting it all together, the completely factored form of the original polynomial is \(-x(x - 3)(x - 8)\) (or we can also write it as \(x(-x + 3)(-x + 8)\) or \(x(3 - x)(8 - x)\), but the first form is more standard when factoring out a negative GCF first).

Answer:

\(-x(x - 3)(x - 8)\) (or equivalent forms like \(x(3 - x)(8 - x)\))