91Ó°ÊÓ

A quadratic expression is a polynomial of the form ax^2 + bx + c. In our exercise, we see such an expression with two different variables: x and y. Recognizing the structure of a quadratic trinomial is the first step. The given problem is \(x^{2}+4xy+3y^{2}\), where:

• \(a = 1\)
• \(b = 4y\)
• \(c = 3y^{2}\)

Like all quadratic expressions, our goal is typically to factor them. Factoring transforms a quadratic trinomial into the product of two binomials. This makes it easier to solve equations or further manipulate expressions.

Factoring by grouping is a method that helps break down polynomials by grouping terms with common factors. Let's understand the process with the following steps:
Step 1: Identify the quadratic expression.
Given expression: \(x^{2}+4xy+3y^{2}\).

Step 2: Find two numbers that multiply to the constant term (3y^2) and add to the middle coefficient (4y). In this case, those numbers are 3y and y because:

• \(3y \times y = 3y^2\)
• \(3y + y = 4y\)

Step 3: Rewrite the middle term using the numbers found:
\(x^{2}+4xy+3y^{2} = x^{2} + 3xy + yx + 3y^{2}\)
Step 4: Group and factor:
\(x^{2} + 3xy + yx + 3y^{2} = x(x + 3y) + y(x + 3y)\)
Step 5: Factor the common binomial:
\(x(x + 3y) + y(x + 3y) = (x + y)(x + 3y)\).
By following these steps, you can make quadratic trinomials much more manageable.

Polynomial factorization involves expressing a polynomial as the product of simpler polynomials. Here, we focus on quadratic polynomials.

• First, recognize if the polynomial is quadratic.
• Use techniques like finding numbers that multiply to the constant term and add to the coefficient of the middle term (important for quadratics).
• Utilize methods like factoring by grouping or other appropriate factoring methods depending on the polynomial.
• Simplify the polynomial into binomials for easier further manipulation.

For our given problem, after grouping, we obtained:
\(x^{2}+4 x y+3 y^{2}=x(x + 3y) + y(x + 3y)\).
Factoring out the common binomial, we get:
\((x + y)(x + 3y)\).
Thus, factorization helps hugely in simplifying and solving polynomial equations efficiently.