Factor the expression completely: 70x^4 + 80x^2.
Understand the Problem
The question is asking to factor the polynomial expression completely, which involves identifying and extracting the common factors from the terms provided.
Answer
The completely factored form is $10x^2(7x^2 + 8)$.
Answer for screen readers
The completely factored form of the polynomial is $$ 10x^2(7x^2 + 8). $$
Steps to Solve
- Identify the Greatest Common Factor (GCF)
First, find the GCF of the coefficients 70 and 80. The GCF is 10. Next, for the variable part, the lowest power of $x$ is $x^2$. Thus, the overall GCF of the expression is $10x^2$.
- Factor out the GCF from each term
Now, we will factor out the GCF $10x^2$ from the expression: $$ 70x^4 + 80x^2 = 10x^2(7x^2 + 8) $$
- Check if the remaining expression can be factored further
Next, we need to check if the expression inside the parentheses, $7x^2 + 8$, can be factored further. Since it is a sum of two squares, it cannot be factored over the real numbers.
The completely factored form of the polynomial is $$ 10x^2(7x^2 + 8). $$
More Information
Factoring helps simplify expressions and solves various algebraic equations. In this example, we found the GCF and factored it out, noting that not all expressions can be further simplified.
Tips
- Forgetting to factor out the GCF completely.
- Assuming that the sum of squares can always be factored, which is not the case.
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