Determine the factors of [tex]$15 x^2+3 x y+10 x+2 y$[/tex].
A. [tex]$(3 x+2)(5 x+y)$[/tex]
B. [tex]$(5 x+y)(2 x+3)$[/tex]
C. [tex]$(3 x+y)(5 x+2)$[/tex]
D. [tex]$(5 x+3)(2 x+y)$[/tex]
Answer (2)
Group the terms: ( 15 x 2 + 3 x y ) + ( 10 x + 2 y ) .
Factor out the GCF from each group: 3 x ( 5 x + y ) + 2 ( 5 x + y ) .
Factor out the common binomial factor: ( 3 x + 2 ) ( 5 x + y ) .
The factors are ( 3 x + 2 ) ( 5 x + y ) .
Explanation
Understanding the Problem We are asked to factor the expression 15 x 2 + 3 x y + 10 x + 2 y . We will use factoring by grouping to achieve this.
Grouping Terms First, we group the terms: ( 15 x 2 + 3 x y ) + ( 10 x + 2 y ) .
Factoring out GCF Next, we factor out the greatest common factor (GCF) from each group. From the first group, 15 x 2 + 3 x y , the GCF is 3 x . Factoring this out, we get 3 x ( 5 x + y ) . From the second group, 10 x + 2 y , the GCF is 2 . Factoring this out, we get 2 ( 5 x + y ) . So now we have 3 x ( 5 x + y ) + 2 ( 5 x + y ) .
Factoring out Common Binomial Now, we notice that ( 5 x + y ) is a common binomial factor. We factor this out to get ( 3 x + 2 ) ( 5 x + y ) .
Final Answer Therefore, the factors of 15 x 2 + 3 x y + 10 x + 2 y are ( 3 x + 2 ) ( 5 x + y ) .
Examples
Factoring expressions like this can help in simplifying complex algebraic equations, which is useful in fields like engineering for designing structures or in economics for modeling market behavior. For example, if you are designing a rectangular garden and know the area can be expressed as 15 x 2 + 3 x y + 10 x + 2 y , factoring it into ( 3 x + 2 ) ( 5 x + y ) helps you determine possible dimensions of the garden in terms of x and y . This allows you to optimize the layout based on available space and resources.
The expression 15 x 2 + 3 x y + 10 x + 2 y factors to ( 3 x + 2 ) ( 5 x + y ) . Thus, the correct answer is option A. This result was obtained through grouping and factoring by common terms.
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