Problem 113 - Algebra and Trigonometry

Question

Factor completely. $$2 x^{2}-7 x y^{2}+3 y^{4}$$

Step-by-Step Solution

Answer

The expression \(2 x^{2}-7 x y^{2}+3 y^{4}\) cannot be factored further since it’s a prime polynomial.

Step 1: Identifying the Form

Identify the form of the expression. In this case, the expression \(2 x^{2}-7 x y^{2}+3 y^{4}\) is a trinomial, a polynomial having three terms.

Step 2: Identifying Possible Factor Pairs

Find pairs of factors of the first and last term whose sum or difference provide the coefficient of the middle term. The factor pairs of \(2x^{2}\) are \((2x, x)\) and the factor of \(3y^{4}\) is \((y^{2},3y^{2})\).

Step 3: Factoring

Under normal circumstances, we would try to identify the factors that meet above conditions. But the factor pairs here do not result in a value that is equivalent to the coefficient of the middle term which is \(-7xy^{2}\). Thus, the trinomial cannot be factored using integer coefficients. The trinomial is a prime polynomial, so it cannot be factored further.