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Chapter 6: Problem 13

Factor each expression. $$4 x+28$$

Short Answer

Expert verified
The factored form of the expression \(4x + 28\) is \(4(x + 7)\).

Step by step solution

01

Identify Common Factor

In the given expression \(4x + 28\), both terms share a common factor of 4. This means that 4 can be factored out.
02

Factor Out the Common Factor

To factor out the common factor, you divide each term in the expression by that common factor. Dividing \(4x\) by 4 you get \(x\), and dividing \(28\) by 4, you get 7. The expression becomes thus: 4(x + 7).
03

Check the Factorization

Verify that the factorization is correct by distributing the 4 back through (multiply the 4 by every term inside the parentheses). 4 times \(x\) gives back \(4x\), and 4 times 7 gives back \(28\). The original expression \(4x + 28\) is thus regained, validating that the factorization is correct.

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that helps us manage and represent relations using symbols and letters. Instead of just dealing with numbers, algebra lets us play with variables. Variables are placeholders for unknowns or quantities that can change.
In our expression, \(4x + 28\), we have a variable \(x\). This variable could stand for any number, making the expression adaptable to different situations.
Using algebra, we can simplify expressions, solve equations, and explore mathematical relationships. This helps in a wide range of fields, from science to engineering.
Finding the Common Factor
The common factor is a number (or variable) that can evenly divide each term in an expression. It's what we "factor out" to simplify the expression.
In our exercise, \(4x + 28\), both terms share a common factor of 4. Recognizing this helps us bring the expression to a simpler form.
Why Look for a Common Factor?
  • Simplifies the expression.
  • Makes solving equations easier.
  • Helps identify relationships between terms.
Factoring out the common factor is a foundational skill in algebra and aids in solving more complex problems.
Applying the Distributive Property
The distributive property is a handy tool in algebra helping us expand or simplify expressions. It states that multiplying a single term across a sum or difference (inside parentheses) can be distributed to each term separately.
For example, in our expression \(4(x + 7)\), 4 should be multiplied by both \(x\) and 7. Thus, we return to the original expression \(4x + 28\) when applied backward.
Benefits of Understanding the Distributive Property:
  • Verifies accuracy of factorization.
  • Makes multiplying expressions simpler.
  • Builds logic for solving complex algebraic equations.
This property not only checks the correctness of our factorization but also supports our ability to manipulate and understand algebraic expressions.

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Most popular questions from this chapter

Set up and solve a proportion. Garden seed is on sale at 3 packets for 70 e. How much will 39 packets cost?

Set up and solve a proportion. A recipe for chocolate chip cookies calls for \(1 \frac{1}{4}\) cups of flour and 1 cup of sugar. The recipe will make \(3 \frac{1}{2}\) dozen cookies. How many cups of flour will be needed to make 12 dozen cookies?

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Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are 0. SEE EXAMPLE 10. (OBJECTIVE 4) $$\frac{x-y}{y-x}$$
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