Questions: Factor completely. 40 x^3+44 x^2-24 x

Transcript text: Factor completely. \[ 40 x^{3}+44 x^{2}-24 x \]

Solution

Solution Steps

To factor the polynomial \(40x^3 + 44x^2 - 24x\) completely, we first look for the greatest common factor (GCF) of the terms. Then, we factor out the GCF and factor the remaining polynomial if possible.

Step 1: Identify the Greatest Common Factor (GCF)

First, we identify the greatest common factor (GCF) of the terms in the polynomial \(40x^3 + 44x^2 - 24x\). The GCF of the coefficients \(40\), \(44\), and \(-24\) is \(4\), and each term contains the variable \(x\). Therefore, the GCF is \(4x\).

Step 2: Factor Out the GCF

Next, we factor out the GCF \(4x\) from the polynomial: \[ 40x^3 + 44x^2 - 24x = 4x(10x^2 + 11x - 6) \]

Step 3: Factor the Quadratic Polynomial

Now, we factor the quadratic polynomial \(10x^2 + 11x - 6\). This can be factored as: \[ 10x^2 + 11x - 6 = (2x + 3)(5x - 2) \]

Step 4: Combine the Factors

Combining the factors, we get the completely factored form of the original polynomial: \[ 40x^3 + 44x^2 - 24x = 4x(2x + 3)(5x - 2) \]