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Chapter 1: Problem 160

Find an equivalent expression by factoring. $$ a b+b $$

Short Answer

Expert verified
The factored expression is \(b(a + 1)\).

Step by step solution

01

Identify the Common Factor

Examine the given expression to identify terms that have a common factor. In the expression \(ab + b\), both terms share the common factor \(b\).
02

Factor Out the Common Factor

Extract the common factor \(b\) from both terms. This can be done by factoring \(b\) out of each term:\[ab + b = b(a + 1)\]
03

Write the Factored Expression

After factoring out the common factor, write the expression in its factored form. The equivalent expression is \(b(a + 1)\).

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Key Concepts

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

Common Factor
When dealing with algebraic expressions, it's important to identify common factors. A common factor is a number or a variable that divides each term in an expression. Seeing common factors can simplify expressions and solve problems more easily.

Let's revisit our expression: a b + b . Here, both terms include the variable 'b'. This means 'b' is a common factor of both terms. Recognizing this common factor is the first step in the process of factoring. Once you spot the common factor, you are on your way to simplifying the expression.
Factoring Techniques
After identifying the common factor in an expression, the next step is to use factoring techniques to simplify it. Factoring techniques involve extracting the common factor from each term.

For the expression: ab + b , we identified 'b' as the common factor. We then factor 'b' out of each term: [ab + b = b(a + 1)] .

Notice how we take 'b' outside of the parentheses and are left with (a + 1) inside. This technique can be applied to more complex expressions as well, breaking them down into simpler terms.
Equivalent Expressions
By factoring an expression, we find an equivalent expression that might be easier to work with. Equivalent expressions are different forms of the same expression. They have the same value but may look different due to transformations like factoring or expanding.

In our example, the original expression a b + b is equivalent to the factored form b(a + 1). Both expressions are equal, meaning they will yield the same result for any value of 'a' and 'b'.

Understanding equivalent expressions is crucial, as it allows flexibility in solving equations and simplifying algebraic problems.

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