Question

write the polynomial as a product of linear factors.\\(x^3 - x^2 - 2x\\)\\(x(x - ?)(x + \quad)\\)

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from the polynomial \(x^3 - x^2 - 2x\). The GCF of \(x^3\), \(-x^2\), and \(-2x\) is \(x\). So we have:
\(x^3 - x^2 - 2x = x(x^2 - x - 2)\)

Step2: Factor the quadratic

Now, we need to factor the quadratic \(x^2 - x - 2\). We look for two numbers that multiply to \(-2\) (the constant term) and add up to \(-1\) (the coefficient of the middle term). The numbers are \(-2\) and \(1\) because \(-2 \times 1 = -2\) and \(-2 + 1 = -1\). So we can factor the quadratic as:
\(x^2 - x - 2 = (x - 2)(x + 1)\)

Step3: Combine the factors

Putting it all together, the polynomial factored as a product of linear factors is:
\(x^3 - x^2 - 2x = x(x - 2)(x + 1)\)

Answer:

The first blank is \(2\) and the second blank is \(1\), so the factored form is \(x(x - 2)(x + 1)\).