Answer :
* Factor out the common term $2x^3$ from the numerator: $2x^9 - 6x^3 = 2x^3(x^6 - 3)$.
* Rewrite the expression as $\frac{2x^3(x^6 - 3)}{2x^3}$.
* Cancel the common factor $2x^3$ from the numerator and the denominator.
* The simplified expression is $\boxed{x^6 - 3}$.
### Explanation
1. Understanding the Problem
We are asked to simplify the expression $\frac{2 x^9-6 x^3}{2 x^3}$. This involves dividing each term in the numerator by the denominator.
2. Factoring the Numerator
First, we can factor out the common term $2x^3$ from the numerator:$$2x^9 - 6x^3 = 2x^3(x^6 - 3)$$
So the expression becomes:
$$\frac{2 x^9-6 x^3}{2 x^3} = \frac{2x^3(x^6 - 3)}{2x^3}$$
3. Canceling Common Factors
Now, we can cancel the common factor $2x^3$ from the numerator and the denominator:
$$\frac{2x^3(x^6 - 3)}{2x^3} = x^6 - 3$$
4. Final Answer
Therefore, the simplified expression is $x^6 - 3$. Comparing this with the given options, we see that option A matches our simplified expression.
### Examples
Simplifying expressions like this is useful in many areas of math and science. For example, if you are calculating the area of a complex shape, you might end up with an expression like the one we simplified. Simplifying it makes it easier to work with and understand. Also, in physics, when dealing with polynomial equations describing motion or forces, simplification is crucial for solving problems efficiently.
* Rewrite the expression as $\frac{2x^3(x^6 - 3)}{2x^3}$.
* Cancel the common factor $2x^3$ from the numerator and the denominator.
* The simplified expression is $\boxed{x^6 - 3}$.
### Explanation
1. Understanding the Problem
We are asked to simplify the expression $\frac{2 x^9-6 x^3}{2 x^3}$. This involves dividing each term in the numerator by the denominator.
2. Factoring the Numerator
First, we can factor out the common term $2x^3$ from the numerator:$$2x^9 - 6x^3 = 2x^3(x^6 - 3)$$
So the expression becomes:
$$\frac{2 x^9-6 x^3}{2 x^3} = \frac{2x^3(x^6 - 3)}{2x^3}$$
3. Canceling Common Factors
Now, we can cancel the common factor $2x^3$ from the numerator and the denominator:
$$\frac{2x^3(x^6 - 3)}{2x^3} = x^6 - 3$$
4. Final Answer
Therefore, the simplified expression is $x^6 - 3$. Comparing this with the given options, we see that option A matches our simplified expression.
### Examples
Simplifying expressions like this is useful in many areas of math and science. For example, if you are calculating the area of a complex shape, you might end up with an expression like the one we simplified. Simplifying it makes it easier to work with and understand. Also, in physics, when dealing with polynomial equations describing motion or forces, simplification is crucial for solving problems efficiently.
