Problem 36 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$3 x^{2}+4 x y+y^{2}$$

Step-by-Step Solution

Answer

The factorized form of \(3x^2 +4xy + y^2\) is \((\sqrt{3}x + y)^2\)

Step 1: Identify the form

Observe the form of the trinomial. It resembles the quadratic form \(a^2 x^2 + 2abxy + b^2 y^2\) which is a form of a perfect square trinomial \((ax + by)^2\). Here, we need to identify the values of \(a\), \(b\), \(x\), and \(y\) in order to factor.

Step 2: Compare the given trinomial.

By comparing \(3x^2 +4xy + y^2\) to the form \(a^2 x^2 + 2abxy + b^2 y^2\). We can deduce the following: \(a = \sqrt{3}\), \(b = 1\), \(x = x\), \(y = y\) since \(\sqrt{3}^2 = 3\), \(1^2 = y^2\), \(2*\sqrt{3}*1 = 4\) and \(x = x\), \(y = y\)

Step 3: Factorize the trinomial.

Now putting these values into the binomial form \((ax+by)^2\), the trinomial can be factorised to: \((\sqrt{3}x + y)^2\)