Problem 41 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 5 c^{2}+20 c+20 $$

Step-by-Step Solution

Answer

The factorized form of the equation is \(5(c^{2}+4c+4)\)

Step 1: Identify Common Factor

Look at the coefficients of each term in the quadratic expression. Here, the coefficients are 5, 20, and 20. The greatest common factor of these numbers is 5.

Step 2: Factor out Common Factor

Factor the quadratic expression by dividing each term by the greatest common factor. That means \(5c^{2}\) divided by 5, 20c divided by 5, and 20 divided by 5.

Step 3: Simplification

Simplify the expression to give: \(c^{2}+4c+4\).

Key Concepts

Greatest Common Factor

Factoring in algebra is similar to breaking down a number into its building blocks. The greatest common factor (GCF) is the highest number that divides exactly into two or more numbers. When we factor a quadratic expression, we want to find the GCF of the coefficients, and if applicable, any variable terms.

For the expression \( 5c^2+20c+20 \), we look at each coefficient: 5, 20, and 20. By analyzing these numbers, we recognize that 5 is the largest number that divides evenly into all of them. Factoring out the GCF is the first step in simplifying the expression. This step is critical because it reduces the complexity of the polynomial, making it easier to work with in subsequent operations such as further factoring or solving the equation.

Simplify Algebraic Expressions

To simplify algebraic expressions, we often start by factoring to reduce the expression to its least complex form. After finding the GCF, we divide each term of the quadratic expression by this factor. This process not only simplifies the calculation but also reveals potential patterns in the expression that can be critical for further simplification or solution techniques, like completing the square or using the quadratic formula.

In our case, after factoring out the GCF of 5, we get \( c^2+4c+4 \) which is a much simpler form of the original quadratic expression. Simplification is key to understanding and solving algebraic problems as it peels away the layers of complexity, revealing the core elements of the expression.

Coefficient Analysis

Closely related to factoring is the coefficient analysis. This involves examining the numerical factors in front of the variable terms in an algebraic expression. Coefficients provide important clues about the behavior of the expression, like indicating how it will open if it's graphed or predicting the number and types of roots the equation might have.

By analyzing the coefficients in the simplified expression \( c^2+4c+4 \), we observe that they form a perfect square trinomial, suggesting that the expression could be further factored. Coefficient analysis helps steer the factoring process by informing which methods to use or properties to apply, such as recognizing patterns that fit certain factorization formulas.