Factor the expression completely: x^3y^5 + x^5y^5.
Understand the Problem
The question is asking to factor the given algebraic expression, which consists of two terms: x³y⁵ + x⁵y⁵. This involves finding the common factors of the terms and expressing them in a simplified form.
Answer
The completely factored expression is $x^3y^5(1 + x^2)$.
Answer for screen readers
The completely factored expression is $x^3y^5(1 + x^2)$.
Steps to Solve
- Identify Common Factors
In the given expression, $x^3y^5 + x^5y^5$, we look for common factors in each term. The factors are:
- For $x^3$ and $x^5$, the common factor is $x^3$ (the lower power).
- For $y^5$ in both terms, it is the same, so it is also included.
- Factor Out the Common Factors
Now we factor out the common factors $x^3y^5$ from each term: $$ x^3y^5(x^0 + x^2) $$ As $x^0$ equals $1$, this simplifies to: $$ x^3y^5(1 + x^2) $$
- Write the Final Factored Expression
The final expression, after factoring out the common factors, is: $$ x^3y^5(1 + x^2) $$
The completely factored expression is $x^3y^5(1 + x^2)$.
More Information
Factoring is a useful algebraic skill that helps to simplify expressions and solve equations. In this case, we found the greatest common factor, which is a fundamental step in algebra. The expression $1 + x^2$ cannot be factored further over the real numbers.
Tips
- Forgetting to include all common factors, such as both parts ($x$ and $y$).
- Not simplifying terms correctly after factoring out the common factors.
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