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Chapter 11: Problem 34

Factor. Check by multiplying. $$ 7 x+28 $$

Short Answer

Expert verified
\[7(x + 4)\]

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look for the greatest common factor (GCF) of the terms in the expression. The terms in this expression are 7 and 28. Determine the GCF of 7 and 28.
02

Factor Out the GCF

The GCF of 7 and 28 is 7. Factor 7 out of each term in the expression. This gives:o\[7x + 28 = 7(x + 4)\]
03

Verify by Multiplying

To verify the factorization, multiply the factored form back out to see if it equals the original expression. Multiply 7 by each term inside the parentheses:\[7(x + 4) = 7x + 28\]Since the expanded form matches the original expression, the factorization is correct.

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor, or GCF, is important in simplifying expressions. It is the largest number that evenly divides all the terms in an expression. In our example, the expression to factor is 7x + 28. We look at the numbers 7 and 28. Checking their factors:
  • Factors of 7: 1, 7
  • Factors of 28: 1, 2, 4, 7, 14, 28
The highest common factor is 7. By factoring out the GCF, we simplify the expression and make it easier to work with.
Verification by Multiplication
After factoring an expression, it is crucial to verify by multiplication. This step ensures the factorization is correct. In our example, we factored 7x + 28 to get 7(x + 4). To verify, we multiply 7 by each term inside the parentheses:
  • 7(x) = 7x
  • 7(4) = 28
Combine these results to get 7x + 28, which matches the original expression. This verification confirms the factorization process is accurate and complete.
Factoring Linear Expressions
Factoring linear expressions is about simplifying them by finding common factors. The goal is to rewrite the expression in a product form. For the expression 7x + 28, we first find the GCF, which is 7. Then, we divide each term by this GCF:
  • 7x 梅 7 = x
  • 28 梅 7 = 4
Thus, the expression simplifies to 7(x + 4).

Factoring aids in managing equations and solving algebraic problems more efficiently. Always ensure you verify your factorization result with multiplication to confirm its correctness.

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