Question
23) \( 2 x y-8 x^{2}+7 y^{3}-28 y^{2} x \)
Ask by Beck Pope. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \(2xy - 8x^{2} + 7y^{3} - 28xy^{2}\) can be factored as \((y - 4x)(2x + 7y^{2})\).
Solution
1. Write the expression by grouping similar terms:
\[
2xy - 8x^2 + 7y^3 - 28xy^2 = (2xy - 8x^2) + (7y^3 - 28xy^2)
\]
2. Factor the greatest common factor (GCF) from each group:
- In the first group, \(2xy - 8x^2\), factor out \(2x\):
\[
2xy - 8x^2 = 2x(y - 4x)
\]
- In the second group, \(7y^3 - 28xy^2\), factor out \(7y^2\):
\[
7y^3 - 28xy^2 = 7y^2(y - 4x)
\]
3. Now the expression is:
\[
2x(y - 4x) + 7y^2(y - 4x)
\]
4. Factor out the common binomial factor \((y - 4x)\):
\[
2x(y - 4x) + 7y^2(y - 4x) = (y - 4x)(2x + 7y^2)
\]
Thus, the fully factored form of the expression is:
\[
(y - 4x)(2x + 7y^2)
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Extra Insights
Let’s break this down! First, we can factor out the common terms from the polynomial. Noticing that \( 2xy \) and \( 8x^2 \) share \( 2x \), and \( 7y^3 \) and \( -28y^2x \) share \( 7y^2 \), we can rewrite the expression as \( 2x(y-4x) + 7y^2(y-4x) \). From there, we can collect like terms: \( (2x + 7y^2)(y - 4x) \). Voila! You've successfully factored the polynomial into its component expressions. Think of polynomials as like pizza! Just as you can slice a pizza to share with friends, you can factor polynomials to make them easier to handle. Each term is a "slice", and factoring allows you to see how they work together to create the whole!
