Question

factor the following polynomial.
$3x^3 + 5x^2 - 2x$
$?(x - )(x + )$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(3x^3\), \(5x^2\), and \(-2x\). The GCF of the coefficients \(3\), \(5\), and \(-2\) is \(1\), and the GCF of the variable terms \(x^3\), \(x^2\), and \(x\) is \(x\). So we factor out \(x\) from the polynomial:
\(3x^3 + 5x^2 - 2x = x(3x^2 + 5x - 2)\)

Step2: Factor the quadratic

Now we need to factor the quadratic \(3x^2 + 5x - 2\). We look for two numbers that multiply to \(3\times(-2)= -6\) and add up to \(5\). The numbers are \(6\) and \(-1\) because \(6\times(-1)= -6\) and \(6 + (-1)=5\). We rewrite the middle term using these numbers:
\(3x^2 + 6x - x - 2\)

Step3: Group and factor

Group the first two terms and the last two terms:
\((3x^2 + 6x) + (-x - 2)\)
Factor out the GCF from each group:
\(3x(x + 2) - 1(x + 2)\)
Now we can factor out \((x + 2)\) from both terms:
\((3x - 1)(x + 2)\)

Step4: Combine with the GCF

We had factored out \(x\) earlier, so putting it all together, the factored form of the polynomial is:
\(x(3x - 1)(x + 2)\)

Answer:

\(x(3x - 1)(x + 2)\)