Questions: Factor the given polynomial completely. 6b^4 - 15b^2 - 36

Solution Steps

To factor the given polynomial \(6b^4 - 15b^2 - 36\), we can follow these steps:

1. Identify the greatest common factor (GCF) of the coefficients.
2. Factor out the GCF.
3. Factor the resulting quadratic expression, if possible.

First, we identify the GCF of the coefficients in the polynomial \(6b^4 - 15b^2 - 36\). The GCF of 6, -15, and -36 is 3.

Next, we factor out the GCF from the polynomial: \[ 6b^4 - 15b^2 - 36 = 3(2b^4 - 5b^2 - 12) \]

Now, we factor the quadratic expression \(2b^4 - 5b^2 - 12\). This can be done by recognizing it as a quadratic in terms of \(b^2\): \[ 2b^4 - 5b^2 - 12 = (2b^2 + 3)(b^2 - 4) \]

We notice that \(b^2 - 4\) is a difference of squares and can be factored further: \[ b^2 - 4 = (b - 2)(b + 2) \]

Combining all the factors, we get: \[ 6b^4 - 15b^2 - 36 = 3(2b^2 + 3)(b - 2)(b + 2) \]

Final Answer