Answer :
- Add the coefficients of the $x^3$ terms: $8 + 6 = 14$, resulting in $14x^3$.
- Add the coefficients of the $x^2$ terms: $9 + (-6) = 3$, resulting in $3x^2$.
- Add the coefficients of the $x$ terms: $0 + (-7) = -7$, resulting in $-7x$.
- Add the constant terms: $19 + (-9) = 10$, resulting in $10$. The final sum is $\boxed{14x^3 + 3x^2 - 7x + 10}$.
### Explanation
1. Understanding the Problem
We are asked to find the sum of two polynomials: $8x^3 + 9x^2 + 19$ and $6x^3 - 6x^2 - 7x - 9$. To do this, we add the coefficients of like terms (terms with the same power of $x$).
2. Adding the $x^3$ Terms
First, let's add the coefficients of the $x^3$ terms: $8 + 6 = 14$. So, the $x^3$ term in the sum is $14x^3$.
3. Adding the $x^2$ Terms
Next, let's add the coefficients of the $x^2$ terms: $9 + (-6) = 9 - 6 = 3$. So, the $x^2$ term in the sum is $3x^2$.
4. Adding the $x$ Terms
Now, let's add the coefficients of the $x$ terms. The first polynomial has no $x$ term, so its coefficient is 0. The second polynomial has a $-7x$ term, so its coefficient is $-7$. Thus, $0 + (-7) = -7$. So, the $x$ term in the sum is $-7x$.
5. Adding the Constant Terms
Finally, let's add the constant terms: $19 + (-9) = 19 - 9 = 10$. So, the constant term in the sum is $10$.
6. Final Sum
Combining all the terms, we get the sum of the two polynomials as $14x^3 + 3x^2 - 7x + 10$.
### Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of electrical circuits. Adding polynomials can help combine different models or analyze the combined effect of different factors. For example, if one polynomial represents the cost of materials and another represents the cost of labor, adding them gives the total cost of a project.
- Add the coefficients of the $x^2$ terms: $9 + (-6) = 3$, resulting in $3x^2$.
- Add the coefficients of the $x$ terms: $0 + (-7) = -7$, resulting in $-7x$.
- Add the constant terms: $19 + (-9) = 10$, resulting in $10$. The final sum is $\boxed{14x^3 + 3x^2 - 7x + 10}$.
### Explanation
1. Understanding the Problem
We are asked to find the sum of two polynomials: $8x^3 + 9x^2 + 19$ and $6x^3 - 6x^2 - 7x - 9$. To do this, we add the coefficients of like terms (terms with the same power of $x$).
2. Adding the $x^3$ Terms
First, let's add the coefficients of the $x^3$ terms: $8 + 6 = 14$. So, the $x^3$ term in the sum is $14x^3$.
3. Adding the $x^2$ Terms
Next, let's add the coefficients of the $x^2$ terms: $9 + (-6) = 9 - 6 = 3$. So, the $x^2$ term in the sum is $3x^2$.
4. Adding the $x$ Terms
Now, let's add the coefficients of the $x$ terms. The first polynomial has no $x$ term, so its coefficient is 0. The second polynomial has a $-7x$ term, so its coefficient is $-7$. Thus, $0 + (-7) = -7$. So, the $x$ term in the sum is $-7x$.
5. Adding the Constant Terms
Finally, let's add the constant terms: $19 + (-9) = 19 - 9 = 10$. So, the constant term in the sum is $10$.
6. Final Sum
Combining all the terms, we get the sum of the two polynomials as $14x^3 + 3x^2 - 7x + 10$.
### Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of electrical circuits. Adding polynomials can help combine different models or analyze the combined effect of different factors. For example, if one polynomial represents the cost of materials and another represents the cost of labor, adding them gives the total cost of a project.
