Problem 51 - Elementary and Intermediate Algebra

Question

Factor. See Example 7 or Example \(12 .\) $$5 a^{2}-25 a+30$$

Step-by-Step Solution

Answer

The factored form is \( 5(a - 2)(a - 3) \).

Step 1: Identify the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of all the terms in the expression. Here, the numbers 5, 25, and 30 all have a GCF of 5. Each term can also be factored by this GCF.

Step 2: Factor out the GCF

Divide each term in the expression by the GCF found in Step 1. When we factor out 5, the expression becomes: \[ 5(a^2 - 5a + 6) \]

Step 3: Factor the Quadratic Expression

Now, factor the quadratic expression inside the parentheses: \( a^2 - 5a + 6 \). To do this, find two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of the middle term). These numbers are -2 and -3.

Step 4: Rewrite and Factor the Quadratic

Rewrite the quadratic expression using the numbers found in Step 3: \[ a^2 - 5a + 6 = (a - 2)(a - 3) \]Now substitute this back into the expression with the GCF factored out: \[ 5(a - 2)(a - 3) \]

Step 5: Simplify and Verify

The expression is now factored. To ensure the factorization is correct, you can expand \( (a - 2)(a - 3) \) to see if it equals \( a^2 - 5a + 6 \). Multiplying \( (a - 2)(a - 3) \) yields \( a^2 - 3a - 2a + 6 = a^2 - 5a + 6 \), confirming our factors are correct.

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, or GCF, is like finding a common friend that can introduce all the terms in a polynomial. It's the largest factor that divides each term without leaving a remainder.
To find the GCF:

• First, break down each number into its prime factors.
• Identify the smallest power of common factors.

For instance, consider the expression provided: 5, 25, and 30. Breaking these down into prime factors gives us:

• 5 = 5
• 25 = 5 × 5
• 30 = 2 × 3 × 5

The common factor here is 5, which is the GCF. Once determined, this GCF can be factored out of the entire polynomial, making the process of simplifying the expression much easier.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, which means the highest exponent of the variable is 2. Typically, these expressions take the form:

• \( ax^2 + bx + c \)

where:

• \( a \) is the coefficient of the square term,
• \( b \) is the coefficient of the linear term, and
• \( c \) is the constant term.

In our example, after factoring out the GCF, we're left with the quadratic \( a^2 - 5a + 6 \).
Quadratics are everywhere, from physics formulas to business models; understanding them is crucial. Here, it’s the structure of the quadratic that allows us to factor it further by identifying sums and products of numbers derived from its terms.

Factoring Quadratics

Factoring a quadratic expression is about breaking it down into simpler components.
The goal is to express the quadratic in the form of two binomials.
To do this:

• Look for two numbers that multiply to \( c \) (the constant term) and add to \( b \) (the coefficient of the linear term).
• In \( a^2 - 5a + 6 \), we find numbers -2 and -3 that multiply to 6 and sum to -5.

Thus, we can express the quadratic as \( (a - 2)(a - 3) \).
This step is not just about numbers but seeing patterns: pairs that work like puzzle pieces to express the same value differently. Once factored, you can always expand to verify, which means multiplying the binomials to check your work. It's a reliable way to ensure the factors are correct and simplifies expressions for further mathematical operations or problem-solving.