Answer :
- Add the coefficients of like terms: $(7x^6 + 3x^6) + (-6x^3) + (10x^2) + (-10 + 4)$.
- Simplify the $x^6$ terms: $7x^6 + 3x^6 = 10x^6$.
- Simplify the constant terms: $-10 + 4 = -6$.
- Combine all terms to get the final answer: $\boxed{10 x^6-6 x^3+10 x^2-6}$.
### Explanation
1. Understanding the problem
We are given two polynomials: $(7x^6 + 10x^2 - 10)$ and $(3x^6 - 6x^3 + 4)$. Our goal is to add these two polynomials together.
2. Adding like terms
To add the polynomials, we combine like terms. This means we add the coefficients of terms with the same power of $x$.
3. Combining the polynomials
So, we have:
$(7x^6 + 10x^2 - 10) + (3x^6 - 6x^3 + 4) = (7x^6 + 3x^6) + (-6x^3) + (10x^2) + (-10 + 4)$
4. Simplifying the expression
Now, we simplify the expression:
$7x^6 + 3x^6 = 10x^6$
$-6x^3$ remains as is.
$10x^2$ remains as is.
$-10 + 4 = -6$
So, the simplified expression is $10x^6 - 6x^3 + 10x^2 - 6$.
5. Finding the correct answer
Comparing our result with the given options, we find that the correct answer is $10x^6 - 6x^3 + 10x^2 - 6$.
6. Final Answer
Therefore, the sum of the two polynomials is $10x^6 - 6x^3 + 10x^2 - 6$.
### Examples
Polynomial addition is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers might use polynomials to model the load distribution and then add these polynomials to find the total load on the bridge. Similarly, in computer graphics, polynomial addition can be used to combine different curves or surfaces to create complex shapes.
- Simplify the $x^6$ terms: $7x^6 + 3x^6 = 10x^6$.
- Simplify the constant terms: $-10 + 4 = -6$.
- Combine all terms to get the final answer: $\boxed{10 x^6-6 x^3+10 x^2-6}$.
### Explanation
1. Understanding the problem
We are given two polynomials: $(7x^6 + 10x^2 - 10)$ and $(3x^6 - 6x^3 + 4)$. Our goal is to add these two polynomials together.
2. Adding like terms
To add the polynomials, we combine like terms. This means we add the coefficients of terms with the same power of $x$.
3. Combining the polynomials
So, we have:
$(7x^6 + 10x^2 - 10) + (3x^6 - 6x^3 + 4) = (7x^6 + 3x^6) + (-6x^3) + (10x^2) + (-10 + 4)$
4. Simplifying the expression
Now, we simplify the expression:
$7x^6 + 3x^6 = 10x^6$
$-6x^3$ remains as is.
$10x^2$ remains as is.
$-10 + 4 = -6$
So, the simplified expression is $10x^6 - 6x^3 + 10x^2 - 6$.
5. Finding the correct answer
Comparing our result with the given options, we find that the correct answer is $10x^6 - 6x^3 + 10x^2 - 6$.
6. Final Answer
Therefore, the sum of the two polynomials is $10x^6 - 6x^3 + 10x^2 - 6$.
### Examples
Polynomial addition is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers might use polynomials to model the load distribution and then add these polynomials to find the total load on the bridge. Similarly, in computer graphics, polynomial addition can be used to combine different curves or surfaces to create complex shapes.
