Question

problem 3 factoring out a monomial got it? a. what is the factored form of $9x^6 + 15x^4 + 12x^2$? b. reasoning what is $-6x^4 - 18x^3 - 12x^2$ written as the product of a polynomial with positive coefficients and a monomial? think how can you check to see if the factored form is correct?

Explanation:

Part a

Step1: Find GCF of coefficients and variables

For coefficients \(9\), \(15\), \(12\), GCF is \(3\). For variables \(x^6\), \(x^4\), \(x^2\), GCF is \(x^2\). So GCF monomial is \(3x^2\).

Step2: Divide each term by GCF

\(\frac{9x^6}{3x^2} = 3x^4\), \(\frac{15x^4}{3x^2} = 5x^2\), \(\frac{12x^2}{3x^2} = 4\).

Step3: Write factored form

\(9x^6 + 15x^4 + 12x^2 = 3x^2(3x^4 + 5x^2 + 4)\)

Step1: Factor out negative GCF

First, find GCF of \(-6x^4\), \(-18x^3\), \(-12x^2\). GCF of coefficients \(6\), \(18\), \(12\) is \(6\), and GCF of variables \(x^4\), \(x^3\), \(x^2\) is \(x^2\). Since we want positive coefficients inside, factor out \(-6x^2\).

Step2: Divide each term by \(-6x^2\)

\(\frac{-6x^4}{-6x^2} = x^2\), \(\frac{-18x^3}{-6x^2} = 3x\), \(\frac{-12x^2}{-6x^2} = 2\).

Step3: Write the product

\(-6x^4 - 18x^3 - 12x^2 = -6x^2(x^2 + 3x + 2)\)

Answer:

\(3x^2(3x^4 + 5x^2 + 4)\)

Part b