Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Factor out the common factor.\((x-1)^{2}+6(x-1)\)

Step-by-Step Solution

Answer

The expression \((x-1)^{2}+6(x-1)\) can be factored out to \((x-1)(x+5)\)

Step 1: Identify the Common Factor

The first step is to identify the common factor in the expression. Here, the common factor is \((x-1)\) because it appears in both terms.

Step 2: Rewrite the Expression as a Product

The next step is to rewrite the expression as a product of the common factor and the remaining factors. To do this, factor out \((x-1)\) from both terms. This gives us: \((x-1)\left((x-1)+6\right)\)

Step 3: Simplify the Expression

The final step is to simplify the expression inside the bracket by performing the addition operation. This gives us the final factored form: \((x-1)(x+5)\)

Key Concepts

Common Factor

A common factor is a value that is present in each term of an algebraic expression. In our exercise, identifying the common factor is the key to simplifying the expression. The expression

• \((x-1)^2 + 6(x-1)\)

includes the term \((x-1)\) in both parts.
When you identify this, you're determining what can be factored out to simplify the expression.

Think of it like finding the shared ingredient in a recipe. If two dishes use chicken, chicken is the common factor. Similarly, in algebra, we look for shared numbers or variables in terms. Recognizing these common factors is crucial. It allows you to rewrite complex expressions more simply.

Expression Simplification

Expression simplification involves making algebraic expressions more manageable by reducing them to a simpler form. Once we've identified the common factor, as we did with \((x-1)\) in our problem, we factor it out.

After factoring out, what remains should also be simplified if possible. The given expression

• \((x-1)((x-1) + 6)\)

allows you to simplify further to \((x-1)(x+5)\).
No terms are wasted when you simplify; you're just making everything clearer and easier to interpret. This not only makes solving easier but also helps when checking your work later. Simplifying is about making everything as clean and direct as possible.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations that form meaningful mathematical phrases. These expressions can represent real-world situations and need manipulation for problem-solving. In our exercise, the expression

• \((x-1)^2 + 6(x-1)\)

is broken into terms: products and sums with variables.

This involves understanding several core operations like addition, subtraction, multiplication, and factoring. By applying properties like distributive property while factoring or simplifying, we transform expressions into different forms.
Understanding these transformations allows you to see the relationships between different parts of the expression and solve equations more efficiently. Practicing with algebraic expressions helps develop problem-solving skills in math!