Question

question
factor completely:
$40x + 2x^2 - 2x^3$
answer attempt 1 out of 2

Explanation:

Step1: Rearrange the polynomial

First, we rearrange the given polynomial \(40x + 2x^{2}-2x^{3}\) in descending order of powers of \(x\). So we get \(- 2x^{3}+2x^{2}+40x\).

Step2: Factor out the greatest common factor (GCF)

The GCF of the terms \(-2x^{3}\), \(2x^{2}\) and \(40x\) is \(- 2x\) (we can also factor out \(2x\) but factoring out \(-2x\) will make the leading coefficient of the quadratic factor positive).
Factoring out \(-2x\) from \(-2x^{3}+2x^{2}+40x\), we have:

$$
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$$

Step3: Factor the quadratic

Now we factor the quadratic \(x^{2}-x - 20\). We need two numbers that multiply to \(- 20\) and add up to \(-1\). The numbers are \(-5\) and \(4\) since \((-5)\times4=-20\) and \(-5 + 4=-1\).
So, \(x^{2}-x - 20=(x - 5)(x + 4)\)

Step4: Write the completely factored form

Substituting the factored form of the quadratic back into the expression, we get:
\(-2x(x - 5)(x + 4)\) or we can also write it as \(2x(-x + 5)(x + 4)\) (but the standard form is usually with the leading coefficient of the quadratic positive, so \(-2x(x - 5)(x + 4)\) is also correct. Another way is to factor out \(2x\) instead of \(-2x\) initially:
If we factor out \(2x\) from \(40x + 2x^{2}-2x^{3}\) (which is \( - 2x^{3}+2x^{2}+40x\)), we get \(2x(-x^{2}+x + 20)\) and then factoring the quadratic inside: \(-x^{2}+x + 20=-(x^{2}-x - 20)=-(x - 5)(x + 4)\), so \(2x\times-(x - 5)(x + 4)=-2x(x - 5)(x + 4)\) which is the same as before.

Answer:

\(-2x(x - 5)(x + 4)\) (or \(2x(5 - x)(x + 4)\) or other equivalent factored forms)