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Chapter 2: Problem 16

Rewrite the expression by taking out the common factors. \(5 x(x+1)+7(x+1)\)

Short Answer

Expert verified
Answer: The simplified form of the expression is \((x+1)(5x+7)\).

Step by step solution

01

Identify the common factor

In the given expression \(5x(x+1)+7(x+1)\), we can see that \((x+1)\) is the common factor in both terms.
02

Factor out the common factor

Now, we will factor out the common factor \((x+1)\) from the expression. To do this, we will rewrite the expression as a product of \((x+1)\) and the sum of the remaining factors: \((x+1)(5x+7)\).
03

Simplify the expression

The simplified expression after factoring out the common factor \((x+1)\) is \((x+1)(5x+7)\). This is the final simplified form of the given expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
A common factor is an expression or number that is shared by terms in an algebraic expression. Identifying common factors is a crucial skill in algebra. It helps in simplifying expressions and solving equations more easily. In an equation, the common factor is a piece of the expression that is found in more than one term.
To find a common factor:
  • Look at each term and identify any similarities.
  • Check for constants, variables, or even expressions that repeat.
  • Factor out the common expression, which means you rewrite the expression as a product.
In our exercise, the expression \(5x(x+1)+7(x+1)\) shares the factor \((x+1)\) in both terms. Recognizing this is key to simplifying the expression later.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, constants, and operators like addition and multiplication. Expressions can take various forms and understanding how to manipulate them is essential in algebra.
Elements of an algebraic expression include:
  • Variables: Symbols that represent unknown values, such as \(x\).
  • Constants: Fixed numbers that have a specific value.
  • Terms: Single components of an expression, like \(5x(x+1)\).
  • Operators: Symbols that represent mathematical operations (e.g., +, -, 脳).
When working with algebraic expressions like \(5x(x+1)+7(x+1)\), being able to identify and understand these components can help simplify and solve algebraic problems effectively.
Simplification Process
Simplifying an algebraic expression involves reducing it to its simplest form. By doing this, we make the expression easier to work with and understand. The process often uses factoring to reach a clearer form.
Steps to simplify an expression:
  • Identify any common factors in the expression.
  • Factor out the common factors from the expression.
  • Rewrite the expression using the factored form.
In the exercise provided, the expression \(5x(x+1)+7(x+1)\) is simplified to \((x+1)(5x+7)\). Removing the common factors allowed us to write the expression more concisely, making it easier to interpret and use in further calculations.

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Most popular questions from this chapter

A car travels 200 miles in \(t\) hours at a speed of \(r\) mph. If the car travels half as fast but three times as long, how far does it travel? (Use the fact that distance equals speed times time.)

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