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Chapter 11: Problem 35

Factor. Check by multiplying. $$ 14 x+21 y $$

Short Answer

Expert verified
The factored expression is 7(2x + 3y), verified by multiplying back to get 14x + 21y.

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Find the greatest common factor (GCF) of the coefficients 14 and 21. The GCF of 14 and 21 is 7 because 7 is the highest number that divides both 14 and 21 evenly.
02

Factor Out the GCF

Factor out 7 from each term in the expression. This means rewriting 14x + 21y as 7(2x) + 7(3y).
03

Write the Factored Expression

Combine the terms inside the parentheses: 7(2x + 3y).
04

Check by Multiplying

Distribute 7 to both terms inside the parentheses to verify the factorization. 7(2x + 3y) equals 7*2x + 7*3y, which simplifies back to 14x + 21y.

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Key Concepts

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

Greatest Common Factor
In algebra, the Greatest Common Factor (GCF) is the largest number that can evenly divide two or more numbers. For example, in the expression \(14x + 21y\), we first look at the coefficients, 14 and 21. To find the GCF:
  • List the factors of each coefficient. The factors of 14 are 1, 2, 7, and 14, while the factors of 21 are 1, 3, 7, and 21.
  • Identify the greatest number common to both lists. Here, that number is 7.
This makes 7 the GCF. Finding the GCF helps simplify expressions by factoring them efficiently.
Factored Expression
A factored expression is the product of a common factor and another expression. In our example \(14x + 21y\), we expressed it in factored form by identifying and pulling out the GCF:
  • The GCF is 7, so we extract 7 from each term: 7(2x) + 7(3y).
  • Combine the terms inside the parentheses to get 7(2x + 3y).
Hence, the factored expression of \(14x + 21y\) is \(7(2x + 3y)\). This expression is simpler and reveals common factors.
Distributive Property
The Distributive Property in algebra states that \(a(b + c) = ab + ac\). It is essential in factoring and simplifying expressions. In our case, we checked our factorization of \(14x + 21y\) by using this property:
  • Starting from \(7(2x + 3y)\), distribute 7 to both terms inside the parentheses.
  • Work out \(7 \times 2x = 14x\) and \(7 \times 3y = 21y\). When you add these terms, you get back the original expression, 14x + 21y.
The Distributive Property ensures that the factorization was done correctly.
Simplifying Expressions
Simplifying expressions makes them easier to work with in algebra. By finding common factors and using the Distributive Property, we can reduce expressions to simpler forms. For \(14x + 21y\):
  • First identify the GCF, which is 7.
  • Factor out 7, rewriting the expression as \(7(2x + 3y)\).
This step-by-step method helps keep the equation manageable, making further calculations straightforward and reducing the probability of errors. Simplifying expressions is a crucial skill in algebra.

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