Problem 89 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$x^{2} y-16 y+32-2 x^{2}$$

Step-by-Step Solution

Answer

The factored form of the polynomial, \(x^{2} y-16 y +32-2 x^{2}\) , is \(-2x^{2}+y(x-4)(x+4)+32\)

Step 1: Gathering Like Terms

Rearrange the polynomial by gathering similar terms together. The rearranged polynomial is: \(-2x^{2}+x^{2}y-16y+32\)

Step 2: Distribute Common Factors

The polynomial, in this case, does not have a common factor in all terms, but a common factor for the second and third terms, which is \(y\), can be taken out. Distribute \(y\) from the second and third terms. The result is: \(-2x^{2}+y(x^{2}-16)+32\)

Step 3: Factor Further if Possible

Now, look to see if any of the resulting terms can be factored further. The second term, \(x^{2}-16\), inside the parenthesis can be factored further because it is a difference of two squares. The difference of squares factors to \(y(x-4)(x+4)+32-2x^{2}\). However, even with this additional factorization, the polynomial as a whole has no other common factors.