Factor completely $5 x(x+3)+6(x+3)$.
Answer (2)
Identify the common factor ( x + 3 ) in both terms.
Factor out the common factor: 5 x ( x + 3 ) + 6 ( x + 3 ) = ( x + 3 ) ( 5 x + 6 ) .
Check if the factors can be factored further. In this case, they cannot.
The completely factored expression is ( x + 3 ) ( 5 x + 6 ) .
Explanation
Understanding the Problem We are asked to factor the expression 5 x ( x + 3 ) + 6 ( x + 3 ) completely. This means we want to rewrite the expression as a product of simpler expressions.
Identifying the Common Factor Notice that both terms in the expression, 5 x ( x + 3 ) and 6 ( x + 3 ) , have a common factor of ( x + 3 ) . We can factor out this common factor.
Factoring out the Common Factor Factoring out ( x + 3 ) from the expression, we get:
5 x ( x + 3 ) + 6 ( x + 3 ) = ( x + 3 ) ( 5 x + 6 )
Checking for Further Factoring Now, we examine the factors ( x + 3 ) and ( 5 x + 6 ) . The factor ( x + 3 ) is a linear expression and cannot be factored further. The factor ( 5 x + 6 ) is also a linear expression and cannot be factored further. Therefore, the expression is completely factored.
Final Answer The completely factored form of the expression is ( x + 3 ) ( 5 x + 6 ) .
Examples
Factoring is a fundamental skill in algebra and is used extensively in solving equations, simplifying expressions, and analyzing functions. For example, if you are designing a rectangular garden and know that the area can be expressed as 5 x 2 + 21 x + 18 , factoring this expression into ( 5 x + 6 ) ( x + 3 ) can help you determine the possible dimensions (length and width) of the garden in terms of x . This allows you to plan the layout of your garden based on the value of x .
The expression 5 x ( x + 3 ) + 6 ( x + 3 ) can be completely factored as ( x + 3 ) ( 5 x + 6 ) . We identified the common factor x + 3 and factored it out. Both factors x + 3 and 5 x + 6 cannot be factored further.
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