Divide: [tex]25 x y^4+60 x y^3[/tex] by [tex]5xy[/tex]
A. [tex]5 x y^3+12 x y^2[/tex]
B. [tex]5 y^2+12[/tex]
C. [tex]5 x y+12 y[/tex]
D. [tex]5 y^3+12 y^2[/tex]
Answer (2)
Divide each term of the expression 25 x y 4 + 60 x y 3 by 5 x y .
Divide 25 x y 4 by 5 x y to get 5 y 3 .
Divide 60 x y 3 by 5 x y to get 12 y 2 .
Combine the results to get the final answer: 5 y 3 + 12 y 2 .
Explanation
Understanding the Problem We are asked to divide the expression 25 x y 4 + 60 x y 3 by 5 x y . This involves dividing each term of the expression by 5 x y and simplifying the result.
Dividing the First Term First, we divide the first term, 25 x y 4 , by 5 x y :
5 x y 25 x y 4 = 5 25 ⋅ x x ⋅ y y 4 = 5 ⋅ 1 ⋅ y 4 − 1 = 5 y 3 . So, 5 x y 25 x y 4 = 5 y 3 .
Dividing the Second Term Next, we divide the second term, 60 x y 3 , by 5 x y :
5 x y 60 x y 3 = 5 60 ⋅ x x ⋅ y y 3 = 12 ⋅ 1 ⋅ y 3 − 1 = 12 y 2 . So, 5 x y 60 x y 3 = 12 y 2 .
Combining the Results Now, we combine the results from the previous two steps: 5 y 3 + 12 y 2 .
Final Answer Therefore, the result of dividing 25 x y 4 + 60 x y 3 by 5 x y is 5 y 3 + 12 y 2 . Looking at the multiple choice options, we see that option D matches our result.
Examples
Understanding how to divide polynomials by monomials is useful in many areas, such as simplifying algebraic expressions, solving equations, and modeling real-world scenarios. For example, if you have a rectangular garden with an area represented by 25 x y 4 + 60 x y 3 and you know one side has a length of 5 x y , you can use polynomial division to find the length of the other side, which would be 5 y 3 + 12 y 2 . This skill is also essential in calculus when dealing with rational functions and their integrals.
To divide 25 x y 4 + 60 x y 3 by 5 x y , we simplify each term separately, resulting in 5 y 3 + 12 y 2 . The correct answer from the choices given is option A: 5 x y 3 + 12 x y 2 .
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