Problem 35 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$9 x^{4}+18 x^{3}+6 x^{2}$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(9x^4 + 18x^3 + 6x^2\) is \(3x^2(3x^2 + 6x + 2)\).

Step 1: Identify the Greatest Common Factor

Observe the polynomial \(9x^4 + 18x^3 + 6x^2\) and identify the greatest common factor. Note that the power of \(x\) in every term is at least 2, and you can take out a constant factor of 3 from every term. Therefore, the greatest common factor is \(3x^2\).

Step 2: Factor out the Greatest Common Factor

Now, divide each term of the polynomial by \(3x^2\) since that's the GCF. So, \(9x^4\) divided by \(3x^2\) is \(3x^2\), \(18x^3\) divided by \(3x^2\) is \(6x\), and \(6x^2\) divided by \(3x^2\) is 2. This gives us a new polynomial which is \(3x^2 + 6x + 2\).

Step 3: Write out the Factorized Polynomial

The polynomial \(9x^4 + 18x^3 + 6x^2\), when factorized, becomes \(3x^2(3x^2 + 6x + 2)\).