Problem 61 - Algebra 2

Question

Factor each expression completely. $$ 3 y^{2}+24 y+45 $$

Step-by-Step Solution

Answer

\(3(y+3)(y+5)\)

Step 1: Determine The Greatest Common Factor (GCF)

Examine the terms in the expression to find the greatest common factor. In the expression \(3 y^{2}+24 y+45\), the greatest common factor is 3.

Step 2: Factor Out the GCF

Factor out the GCF from the expression. This results in \(3(y^{2}+8y+15)\).

Step 3: Factor the Quadratic Expression

Now factor the quadratic expression inside the parentheses. We are looking for two numbers that add up to 8 and multiply to 15. These numbers are 3 and 5. So, we can factor the quadratic as \((y+3)(y+5)\).

Step 4: Write the Fully Factored Expression

Combine the GCF and the factored quadratic expression to obtain the fully factored form: \(3(y+3)(y+5)\).

Key Concepts

Greatest Common Factor (GCF)

Understanding the greatest common factor (GCF) is essential when it comes to factoring algebraic expressions. The GCF is the highest number that divides evenly into all terms of a given expression. For instance, when factoring the quadratic expression 3y^2 + 24y + 45, you first identify the GCF, which for these terms is 3.

Factoring out the GCF simplifies the expression and reduces the complexity of the remaining polynomial, making further steps in the factoring process more manageable. It is akin to decluttering a room before cleaning it in depth - by removing the common items first (the GCF), you can more easily organize what's left (the simplified polynomial).

Factor Quadratics

Once the GCF is factored out, we encounter a quadratic expression that requires further factoring. A quadratic is an algebraic expression of the second degree, generally represented as ax^2+bx+c. To factor quadratics, we look for two binomials that when multiplied together give us back the original quadratic.

The process involves finding two numbers that both add up to the coefficient of the middle term (b) and multiply to the constant term (c) when a is equal to 1. In the given exercise, the quadratic expression was y^2 + 8y + 15. Here, the numbers 3 and 5 fulfill both conditions (adding to 8 and multiplying to 15), hence the quadratic factors to (y + 3)(y + 5).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the context of factoring, understanding the structure of these expressions is crucial. They can be simple, with only one term (monomial), or more complex, like a quadratic expression which has three terms.

Each term in an algebraic expression contributes to its overall behavior and the strategies used to simplify or factor the expression. For example, in 3(y^2 + 8y + 15), we initially considered each term's GCF, then moved to factor a trinomial, which involves terms in ‘square’, ‘linear’, and ‘constant’ forms.

Polynomial Factoring

Polynomial factoring is a process that breaks down a polynomial into the product of its simpler factors. The aim is to represent the expression in a way that makes its properties and solutions more apparent. Factoring polynomials can involve a variety of techniques including finding the GCF, grouping, and using special formulas such as the difference of squares, or trinomial factoring as seen with quadratic expressions.

Mastering polynomial factoring is crucial for solving equations and inequalities, simplifying expressions, and understanding the graphical behavior of polynomial functions. Each factored form provides insights into the roots or solutions of a polynomial equation, which correspond to the x-intercepts of its graph.