Problem 72 Use the distributive law to fact... [FREE SOLUTION]

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Chapter 1: Problem 72

Use the distributive law to factor each of the following. Check by multiplying. $$ 20 a+5 $$

Short Answer

Expert verified

5(4a + 1)

Step by step solution

Identify the Greatest Common Factor (GCF)

Look at the numbers in each term. The terms are 20a and 5. The greatest common factor for 20 and 5 is 5.

Factor out the GCF

Rewrite each term as a product of the GCF and another factor. This gives:\[20a + 5 = 5(4a) + 5(1)\]Now, factor out the GCF (5):\[20a + 5 = 5(4a + 1)\]

Distributive Property Check

Multiply back to check if the factorization is correct:\[5(4a + 1) = 5 \times 4a + 5 \times 1 = 20a + 5\]The original expression is obtained, 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.

Greatest Common Factor (GCF)

To understand factoring, we must first learn about the Greatest Common Factor (GCF). The GCF is the largest number that can evenly divide both of the terms in the expression.
For example, in the expression $20a + 5$, we must find the GCF for the numbers 20 and 5.
The factors of 20 are: 1, 2, 4, 5, 10, 20.
The factors of 5 are: 1, 5.
So, the GCF is the largest number that appears in both lists, which is 5. This means 5 is the GCF for 20 and 5.
Understanding the GCF helps us simplify expressions and solve algebraic problems more easily.

Factoring Expressions

Once we know the GCF, we can use it to factor the expression.
Factoring means breaking down an expression into simpler terms which, when multiplied together, give back the original expression.
Let's look at the expression $20a + 5$. With the GCF identified as 5, we rewrite each term by expressing them as a product of the GCF and another factor:
\[20a = 5 \times 4a\] and \[5 = 5 \times 1\]
Thus, $20a + 5$ can be written as:
\[5 \times 4a + 5 \times 1\]
Now, we can factor out the GCF (5) from both terms, leading to:
\[5(4a + 1)\]
This process is called factoring the expression.

Distributive Property

The Distributive Property is a helpful algebraic property used when factoring expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products.
Mathematically, it is written as:
\[a(b + c) = ab + ac\]
In our expression $5(4a + 1)$, we can apply the distributive property to check our factorization:
\[5 \times 4a + 5 \times 1\]
This results in:
\[20a + 5\]
Which confirms that our original expression, $20a + 5$, matches the factorized form $5(4a + 1)$.
This verification step is essential to ensure our factorization work is correct.

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91影视

Use the distributive law to factor each of the following. Check by multiplying. $$ 20 a+5 $$

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor out the GCF

Distributive Property Check

Key Concepts

Greatest Common Factor (GCF)

Factoring Expressions

Distributive Property

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