Problem 35 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$2 x^{2}+3 x y+y^{2}$$

Step-by-Step Solution

Answer

The factorised form of the trinomial \(2 x^{2}+3 x y+y^{2}\) is \((\sqrt{2}x+y)^2\).

Step 1: Identify the Pattern

The given trinomial follows the pattern \(a^2+2ab+b^2\), which is recognised as a perfect square trinomial. Each term in the trinomial corresponds to a part of this pattern with \(a^2 = 2x^2\), \(2ab = 3xy\), and \(b^2 = y^2\).

Step 2: Determine \(a\) and \(b\)

From the pattern, we determine that \(a = \sqrt{2}x\) and \(b = y\).

Step 3: Factorize the Trinomial

A perfect square trinomial \(a^2+2ab+b^2\) can be factored into \((a+b)^2\). Substituting our determined values of \(a\) and \(b\), the factorised form of the trinomial is \((\sqrt{2}x+y)^2\).