Factor the expression completely: -10x^5 - 5x.
Understand the Problem
The question is asking to factor the polynomial expression completely, which involves identifying common factors and writing it in a factored form.
Answer
The factored expression is $-5x(2x^4 + 1)$.
Answer for screen readers
The completely factored form of the expression is: $$ -5x(2x^4 + 1) $$
Steps to Solve
- Identify Common Factors
First, we need to identify the common factor in the terms of the polynomial. The two terms are $-10x^5$ and $-5x$.
- Extract the Greatest Common Factor (GCF)
The coefficients $-10$ and $-5$ have a GCF of $-5$. The variable part has a common factor of $x$. Therefore, the GCF of the entire expression is $-5x$.
- Factor Out the GCF
Now, we factor out the GCF $-5x$ from the expression: $$ -10x^5 - 5x = -5x(2x^4 + 1) $$
- Check for Further Factorization
Next, we check if the remaining expression $2x^4 + 1$ can be factored further. It does not factor nicely over the real numbers.
The completely factored form of the expression is: $$ -5x(2x^4 + 1) $$
More Information
Factoring polynomials is a fundamental skill in algebra, and identifying the greatest common factor is often the first step in this process. This form, $-5x(2x^4 + 1)$, cannot be further simplified over the real numbers.
Tips
- Not identifying the GCF correctly. Always check coefficients and variable degrees.
- Forgetting to factor out the negative sign, which can lead to incorrect final answers.
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