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Chapter 1: Problem 76

Use the distributive law to factor each of the following. Check by multiplying. $$ 25 y+30 $$

Short Answer

Expert verified
\[25y + 30 = 5(5y + 6)\]

Step by step solution

01

Identify the greatest common factor (GCF)

Look for the largest number that divides both coefficients of the terms in the expression. The coefficients are 25 and 30. The GCF of 25 and 30 is 5.
02

Factor out the GCF

Write the expression as the product of the GCF and another expression. Factor out the 5: \[25y + 30 = 5(5y + 6) \]
03

Check by multiplying

Multiply the factor back to verify the result. \[5(5y + 6) = 5 \times 5y + 5 \times 6 = 25y + 30 \] The original expression matches, so the factorization is correct.

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Key Concepts

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

distributive law
To start, let's understand what the distributive law is. The distributive law helps us to multiply a number by a sum or difference. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. In mathematical terms, for any numbers a, b, and c: a(b + c) = ab + acThis law allows us to break down complex multiplication into simpler steps. When factoring, we often use the distributive law in reverse to simplify expressions by finding common factors and grouping them.
greatest common factor
The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. Let's find the GCF for the coefficients in our given problem: 25 and 30. The factors of 25 are 1, 5, and 25. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The common factors are 1 and 5, and the greatest common factor is 5. Finding the GCF is important because it allows us to simplify or factor expressions, making them easier to work with.
factoring expressions
Factoring expressions involves breaking down an expression into a product of simpler expressions. In our example, the expression is 25y + 30. Using the GCF, we factor out the common factor from each term:25y + 30 = 5(5y + 6) By factoring out the GCF (5), we rewrite the expression as the product of 5 and (5y + 6). To verify the factorization, we can use the distributive law to multiply 5 by each term inside the parentheses:5(5y + 6) = 5 * 5y + 5 * 6 = 25y + 30 The original expression and our factorized expression match, confirming that our factorization is correct. Factoring simplifies expressions and helps solve equations more easily.

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Most popular questions from this chapter

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