Problem 107 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. After factoring \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x,\) I noticed that I factored the monomial \(20 x^{3}\) in two different ways.

Step-by-Step Solution

Answer

The statement 'monomial \(20 x^{3}\) factored in two different ways' makes sense because the initial polynomials are different and require different common factors when factored.

Step 1: Factoring the Monomial from the first polynomial

The first polynomial given is \(20 x^{3}+8 x^{2}\). The common monomial for this polynomial can be factored out and is \(4 x^{2}\). So, factoring the polynomial becomes: \(4x^{2}(5x + 2)\)

Step 2: Factoring the Monomial from the second polynomial

The second polynomial given is \(20 x^{3}+10 x\). The common monomial for this polynomial can be factored out and is \(10x\). So, factoring the polynomial becomes: \(10x(2x^2 + 1)\)

Step 3: Comparing the factored forms

Comparing the factored expression of the first polynomial and second polynomial, it is clear that the monomial \(20x^3\) has different factored forms in two polynomials. This is because the initial polynomials have different terms and therefore require different common factor. Hence, the statement that the monomial \(20x^3\) factored in two different ways makes sense.