Problem 53 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ x^{2}+7 x y+10 y^{2} $$

Step-by-Step Solution

Answer

The trinomial factors to \((x + 5y)(x + 2y)\).

Step 1: Identify the Greatest Common Factor (GCF)

Examine each term in the trinomial \(x^2 + 7xy + 10y^2\). There is no common factor other than 1 for these terms, so there is no need to factor out a GCF here.

Step 2: Set Up Factoring by Pairing

We need to factor \(x^2 + 7xy + 10y^2\) into the form \((ax + by)(cx + dy)\). We look for two numbers that multiply to \(10y^2\) (the last term) and add to \(7y\) (the middle term). These numbers are 5 and 2.

Step 3: Write the Expression as a Pair

Rewrite the middle term as a sum using the numbers found in Step 2: \(x^2 + 5xy + 2xy + 10y^2\).

Step 4: Factor by Grouping

Group the terms into two pairs: \((x^2 + 5xy) + (2xy + 10y^2)\). Factor each group: \(x(x + 5y) + 2y(x + 5y)\). Notice that both groups contain the binomial \((x + 5y)\).

Step 5: Factor Out the Common Binomial

Factor out the common binomial \((x + 5y)\) from the two groups: \((x + 5y)(x + 2y)\). This is the factored form of the trinomial.

Key Concepts

Greatest Common Factor

In algebra, the Greatest Common Factor (GCF) is the largest number, or expression, that divides all terms of a polynomial without leaving a remainder. Identifying the GCF is a crucial initial step when factoring polynomials.

In some cases, factoring out the GCF can simplify the expression, making further steps easier. However, in the trinomial \(x^2 + 7xy + 10y^2\), there isn't a GCF among the terms other than 1. This means there's no need to factor out a GCF before proceeding with other factoring techniques.

Even when it's absent, always checking for a GCF is a best practice when beginning to factor any polynomial.

Factoring by Grouping

Factoring by grouping involves rearranging a polynomial's terms and then grouping them into pairs that can be factored individually. This technique is especially useful in trinomials and polynomials with more than three terms.

For the trinomial \(x^2 + 7xy + 10y^2\), we first split the middle term. The target is to find two numbers that multiply to the last term's coefficient, \(10y^2\), and add to the middle term's coefficient, \(7y\). We identified these as 5 and 2, allowing us to rewrite the polynomial as \(x^2 + 5xy + 2xy + 10y^2\).

Next, we group the terms: \((x^2 + 5xy) + (2xy + 10y^2)\). Each pair is then factored separately, revealing a common binomial factor.

Polynomials

Polynomials are expressions composed of variables and coefficients, involving terms of varying degrees. The standard form arranges terms from highest to lowest degree.
A trinomial is a specific kind of polynomial with exactly three terms. In our exercise, \(x^2 + 7xy + 10y^2\) is a quadratic trinomial, as its highest power of \(x\) is 2. Factoring polynomials like these is essential in solving algebraic problems.
Understanding the structure and order of polynomials helps determine the best factoring methods to use.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They consist of numbers, variables, and operations that define them. Unlike equations, they do not have an equals sign.

The expression in our example, \(x^2 + 7xy + 10y^2\), is an algebraic expression that needs to be factored. Factoring transforms it into a simpler version, making it easier to use in equations.
Understanding how to manipulate these expressions is vital, as it lays the foundation for solving more complex equations in algebra.