$1 a^2+a b+a x+b x=$
Answer (1)
Group the terms: ( a 2 + ab ) + ( a x + b x ) .
Factor out common factors from each group: a ( a + b ) + x ( a + b ) .
Factor out the common binomial factor ( a + b ) : ( a + b ) ( a + x ) .
The factored form of the expression is ( a + b ) ( a + x ) .
Explanation
Understanding the Problem We are given the expression a 2 + ab + a x + b x and asked to factor it. Factoring is the process of breaking down an expression into a product of simpler expressions. In this case, we will use factoring by grouping.
Grouping the Terms First, we group the terms in pairs: ( a 2 + ab ) + ( a x + b x ) .
Factoring out Common Factors Next, we factor out the greatest common factor (GCF) from each group. From the first group ( a 2 + ab ) , the GCF is a . Factoring out a , we get a ( a + b ) . From the second group ( a x + b x ) , the GCF is x . Factoring out x , we get x ( a + b ) . So the expression becomes a ( a + b ) + x ( a + b ) .
Factoring out the Common Binomial Now, we notice that ( a + b ) is a common factor in both terms. We factor out ( a + b ) from the entire expression: ( a + b ) ( a + x ) .
Final Answer Therefore, the factored form of the expression a 2 + ab + a x + b x is ( a + b ) ( a + x ) .
Examples
Factoring is a fundamental concept in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. In finance, factoring can be used to analyze investment portfolios and manage risk. Imagine you are designing a rectangular garden. You know the total area of the garden can be represented by the expression a 2 + ab + a x + b x , where a and b are related to the dimensions. By factoring this expression into ( a + b ) ( a + x ) , you can easily determine the possible dimensions of the garden based on different values of a , b , and x . This helps you plan the layout efficiently and make the best use of the available space.
