Problem 77 Use the distributive law to fact... [FREE SOLUTION]

Chapter 1: Problem 77

Use the distributive law to factor each of the following. Check by multiplying. $$ 3 a+9 b $$

Short Answer

Expert verified

3(a + 3b)

Step by step solution

Identify the Common Factor

Look at each term in the expression and identify the greatest common factor (GCF). For the expression 3a + 9b, the GCF is 3.

Factor Out the GCF

Rewrite the expression by factoring out the GCF. This means you divide each term by the GCF: a) 3a 梅 3 = ab) 9b 梅 3 = 3bThus, the expression becomes 3(a + 3b).

Check by Multiplying

To verify the factorization, distribute the GCF back through the terms inside the parentheses. Multiply:3(a + 3b) = 3*a + 3*3b = 3a + 9bThe original expression is obtained, confirming 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

Understanding the greatest common factor (GCF) is essential when dealing with expressions in algebra. The GCF is the largest number that evenly divides all terms in the expression. For example, in the expression 3a + 9b, we need to find the largest number that can divide both 3a and 9b.

Think of it like breaking down the numbers 3 and 9 into their prime factors:

3 is a prime number and its only factors are 1 and 3.
9 breaks down into 3 * 3.

The common factor here is 3. So, 3 is the GCF that you can factor out from both terms. This process helps simplify the expression and makes it easier to work with, especially in more complex algebraic problems.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions. Using the GCF is a straightforward method to start factoring.

For the expression 3a + 9b, once you've identified that the GCF is 3, you can factor it out by rewriting each term divided by the GCF.

Here's the step-by-step process:

First, 3a divided by 3 gives us a.
Next, 9b divided by 3 gives us 3b.

So, 3a + 9b can be rewritten as 3(a + 3b).

By factoring out the GCF, you simplify the expression and make it more manageable. This technique is not only useful for small numbers but also for more complicated algebraic expressions.

Checking Multiplication

Always double-check your factorization by multiplying the factors back together. This step ensures that you鈥檝e factored the expression correctly.

To verify our factorization of 3(a + 3b), distribute 3 through the terms inside the parentheses:

3 * a = 3a
3 * 3b = 9b

So, 3(a + 3b) indeed multiplies back to 3a + 9b.

When these steps match the original expression, you confirm the factorization is correct. This practice is crucial, especially as expressions become more complicated, ensuring your work is accurate and reliable.

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91影视

Use the distributive law to factor each of the following. Check by multiplying. $$ 3 a+9 b $$

Short Answer

Step by step solution

Identify the Common Factor

Factor Out the GCF

Check by Multiplying

Key Concepts

Greatest Common Factor

Factoring

Checking Multiplication

One App. One Place for Learning.

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