Factor the expression completely. -16x^2 - 6x^4
Understand the Problem
The question is asking to factor the given polynomial expression completely. The expression provided is a polynomial in the variable x, and we need to identify any common factors and factor it accordingly.
Answer
The completely factored form is $-2x^2(8 + 3x^2)$.
Answer for screen readers
The completely factored form of the expression is: $$ -2x^2(8 + 3x^2) $$
Steps to Solve
- Identify common factors
First, we need to identify the common factors in both terms of the polynomial expression $-16x^2 - 6x^4$.
- Factor out the GCF
The greatest common factor (GCF) of $-16$ and $-6$ is $-2$. The lowest power of $x$ present in both terms is $x^2$. Therefore, we can factor out $-2x^2$ from the expression.
- Perform the factoring
Factoring out $-2x^2$, we have: $$ -2x^2(8 + 3x^2) $$
- Check for further factoring
Now we need to check if the remaining expression $8 + 3x^2$ can be factored further. Since it is a sum of squares and does not factor over the reals, we stop here.
The completely factored form of the expression is: $$ -2x^2(8 + 3x^2) $$
More Information
The expression $-2x^2(8 + 3x^2)$ shows that we have factored out the GCF and simplified the polynomial. The term $8 + 3x^2$ is irreducible over the real numbers.
Tips
- Not identifying the GCF: Students may overlook the common factors in the terms.
- Assuming all expressions factor further: Not every polynomial can be factored into simpler polynomial expressions; sums of squares, like $8 + 3x^2$, cannot be factored further in the real number system.
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