Certainly! Let's solve this problem step-by-step.
### Part (a)
We are given the polynomials:
1. [tex]\(3p - 4q + 4r\)[/tex]
2. [tex]\(2p + 3q - 8r\)[/tex]
3. [tex]\(q - 6p + r\)[/tex]
To add these polynomials, we combine the like terms for [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex].
Let's begin by organizing and summing the coefficients for each variable:
1. Terms involving [tex]\(p\)[/tex]:
- First polynomial: [tex]\(3p\)[/tex]
- Second polynomial: [tex]\(2p\)[/tex]
- Third polynomial: [tex]\(-6p\)[/tex]
Summing these, we get:
[tex]\[
3p + 2p - 6p = (3 + 2 - 6)p = -p
\][/tex]
2. Terms involving [tex]\(q\)[/tex]:
- First polynomial: [tex]\(-4q\)[/tex]
- Second polynomial: [tex]\(3q\)[/tex]
- Third polynomial: [tex]\(q\)[/tex]
Summing these, we get:
[tex]\[
-4q + 3q + q = (-4 + 3 + 1)q = 0q = 0
\][/tex]
3. Terms involving [tex]\(r\)[/tex]:
- First polynomial: [tex]\(4r\)[/tex]
- Second polynomial: [tex]\(-8r\)[/tex]
- Third polynomial: [tex]\(r\)[/tex]
Summing these, we get:
[tex]\[
4r - 8r + r = (4 - 8 + 1)r = -3r
\][/tex]
Putting it all together, the sum of the polynomials in part (a) is:
[tex]\[
\boxed{-p - 3r}
\][/tex]
### Part (b)
We are given the polynomials:
1. [tex]\(5a - 8b + 2c\)[/tex]
2. [tex]\(3c - 4b - 2a\)[/tex]
3. [tex]\(6b - c - a\)[/tex]
4. [tex]\(3a - 2c - 3b\)[/tex]
To add these polynomials, we again combine the like terms for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
Let's begin by organizing and summing the coefficients for each variable:
1. Terms involving [tex]\(a\)[/tex]:
- First polynomial: [tex]\(5a\)[/tex]
- Second polynomial: [tex]\(-2a\)[/tex]
- Third polynomial: [tex]\(-a\)[/tex]
- Fourth polynomial: [tex]\(3a\)[/tex]
Summing these, we get:
[tex]\[
5a - 2a - a + 3a = (5 - 2 - 1 + 3)a = 5a
\][/tex]
2. Terms involving [tex]\(b\)[/tex]:
- First polynomial: [tex]\(-8b\)[/tex]
- Second polynomial: [tex]\(-4b\)[/tex]
- Third polynomial: [tex]\(6b\)[/tex]
- Fourth polynomial: [tex]\(-3b\)[/tex]
Summing these, we get:
[tex]\[
-8b - 4b + 6b - 3b = (-8 - 4 + 6 - 3)b = -9b
\][/tex]
3. Terms involving [tex]\(c\)[/tex]:
- First polynomial: [tex]\(2c\)[/tex]
- Second polynomial: [tex]\(3c\)[/tex]
- Third polynomial: [tex]\(-c\)[/tex]
- Fourth polynomial: [tex]\(-2c\)[/tex]
Summing these, we get:
[tex]\[
2c + 3c - c - 2c = (2 + 3 - 1 - 2)c = 2c
\][/tex]
Putting it all together, the sum of the polynomials in part (b) is:
[tex]\[
\boxed{5a - 9b + 2c}
\][/tex]
So, the final results are:
- Part (a): [tex]\(\boxed{-p - 3r}\)[/tex]
- Part (b): [tex]\(\boxed{5a - 9b + 2c}\)[/tex]
