Problem 36 - 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. $$32 x^{4}+2 x^{3}+8 x^{2}$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(32 x^{4}+2 x^{3}+8 x^{2}\) is \(2 x^{2}(16 x^{2}+x+4)\)

Step 1 Identifying the GCF

Look at each term in the given polynomial, \(32 x^{4}+2 x^{3}+8 x^{2}\), find the greatest number that individually divides all coefficients, which are 32, 2, and 8. The greatest common factor is 2 here. Now check the common power of x in all three terms, which is 2. The term having the smallest exponent will be a part of the GCF, it's \(x^{2}\) here.

Step 2 Factoring out the GCF

Divide each term of the polynomial by the GCF \(2 x^{2}\) and write the polynomial as a product of the GCF and the result of the division, the factored form: \(32 x^{4}/ 2 x^{2} + 2 x^{3}/ 2 x^{2} + 8 x^{2}/2 x^{2}\), which simplifies to \(2 x^{2}(16 x^{2}+x+4)\)

Step 3 Final Check

Re-distribute the GCF \(2 x^{2}\) to ensure you get back the original polynomial. Multiply \(2 x^{2}\) into each term in parentheses: \(2 x^{2} * 16x^{2} = 32 x^{4}\), \(2 x^{2} * x = 2 x^{3}\) and \(2 x^{2} * 4 = 8 x^{2}\). So, the original polynomial is obtained, and the factoring is done correctly.