Problem 68 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression \(5y + 5z\) factors to \(5(y + z)\).

Step 1: Identify the common factor

Look at the terms in the expression \(5y + 5z\). Identify the common factor in both terms. Both terms have a common factor of 5.

Step 2: Factor out the common factor

Rewrite the expression by factoring out the common factor of 5. This means writing the expression as 5 times a sum: \[ 5(y + z) \]

Step 3: Check by multiplying

To verify the factorization, distribute 5 back into the terms inside the parenthesis:\[ 5(y + z) = 5y + 5z \]Since the original expression is obtained, the factorization is correct.

Key Concepts

factoring

Factoring is a fundamental concept in algebra. It involves rewriting an expression as a product of its factors. Factoring simplifies expressions and solves polynomial equations more easily.
One common way to factor is by using the distributive law. This method finds a common factor in all terms and rewrites the expression using this factor.
Let's look at the problem: Factor the expression: $$5y + 5z$$. We identify the common factor and rewrite the expression. Next, we discuss the concept of a common factor.

common factor

A common factor is a number or algebraic term that divides each term in an expression without a remainder. Knowing how to find and use a common factor is crucial for simplifying and factoring expressions.
In our example, the expression is $$5y + 5z$$.

• Both terms, 5y and 5z, have a coefficient of 5.
• This coefficient is the common factor.

Thus, we factor out the common factor 5:
$$5y + 5z = 5(y + z)$$. By factoring out the 5, we simplify the expression to $$5(y + z)$$. This process helps to break down complex expressions into simpler components. In the next section, we will explore how factoring can be verified.

verification

Verification ensures the accuracy of our factorization work. It involves checking if the factorized expression, when expanded, returns the original expression.
For the expression we've been working with, let's verify the factorization:

• Start with the factorized form: $$5(y + z)$$
• Apply the distributive law: $$5(y + z) = 5 \times y + 5 \times z = 5y + 5z$$

We see that multiplying the factor out returns the original expression, $$5y + 5z$$. This confirms that our factorization is correct.
Verification helps identify and correct any errors in the factoring process, ensuring accuracy in algebraic manipulations.