Certainly! Let's go through the process of adding the two given polynomials step by step.
We are given two polynomials:
[tex]\[ (8r^2 - 7r - 9) \][/tex]
and
[tex]\[ (-r^2 + r). \][/tex]
### Step-by-Step Solution
1. Identify the like terms:
- The first polynomial: [tex]\(8r^2 - 7r - 9\)[/tex]
- The second polynomial: [tex]\(-r^2 + r\)[/tex]
2. Group the like terms together:
- For [tex]\(r^2\)[/tex] terms: [tex]\(8r^2\)[/tex] from the first polynomial and [tex]\(-r^2\)[/tex] from the second polynomial.
- For [tex]\(r\)[/tex] terms: [tex]\(-7r\)[/tex] from the first polynomial and [tex]\(+r\)[/tex] from the second polynomial.
- For the constant term: [tex]\(-9\)[/tex] from the first polynomial (the second polynomial does not have a constant term).
3. Add the coefficients of the like terms:
- [tex]\(r^2\)[/tex] term: [tex]\(8r^2 + (-r^2) = 8r^2 - r^2 = 7r^2\)[/tex]
- [tex]\(r\)[/tex] term: [tex]\(-7r + r = -7r + r = -6r\)[/tex]
- Constant term: [tex]\(-9 + 0 = -9\)[/tex]
4. Combine the results:
- The expanded polynomial in standard form will be: [tex]\( 7r^2 - 6r - 9 \)[/tex]
Therefore, the final answer is:
[tex]\[ (8r^2 - 7r - 9) + (-r^2 + r) = 7r^2 - 6r - 9 \][/tex]
