Sure! Let's go through each part of the question step-by-step and subtract the second polynomial from the first.
### Part (a):
Consider the polynomials:
[tex]\[ p(x) = 3x^3 + 5x - 4 \][/tex]
[tex]\[ q(x) = 6x^4 - x + 2 \][/tex]
We need to subtract [tex]\( q(x) \)[/tex] from [tex]\( p(x) \)[/tex]:
[tex]\[ p(x) - q(x) = (3x^3 + 5x - 4) - (6x^4 - x + 2) \][/tex]
Expanding the subtraction:
[tex]\[ p(x) - q(x) = 3x^3 + 5x - 4 - 6x^4 + x - 2 \][/tex]
Combining like terms:
[tex]\[ p(x) - q(x) = -6x^4 + 3x^3 + 5x + x - 4 - 2 \][/tex]
[tex]\[ p(x) - q(x) = -6x^4 + 3x^3 + 6x - 6 \][/tex]
So, the resulting polynomial for part (a) is:
[tex]\[ -6x^4 + 3x^3 + 6x - 6 \][/tex]
### Part (b):
Consider the polynomials:
[tex]\[ p(y) = 6y^2 + y - 2 \][/tex]
[tex]\[ q(y) = -8y^3 + 3y^2 + 7 \][/tex]
We need to subtract [tex]\( q(y) \)[/tex] from [tex]\( p(y) \)[/tex]:
[tex]\[ p(y) - q(y) = (6y^2 + y - 2) - (-8y^3 + 3y^2 + 7) \][/tex]
Expanding the subtraction:
[tex]\[ p(y) - q(y) = 6y^2 + y - 2 + 8y^3 - 3y^2 - 7 \][/tex]
Combining like terms:
[tex]\[ p(y) - q(y) = 8y^3 + 6y^2 - 3y^2 + y - 2 - 7 \][/tex]
[tex]\[ p(y) - q(y) = 8y^3 + 3y^2 + y - 9 \][/tex]
So, the resulting polynomial for part (b) is:
[tex]\[ 8y^3 + 3y^2 + y - 9 \][/tex]
### Part (c):
Consider the polynomials:
[tex]\[ p(x) = 8x^3 + 6x^2 + 5x + 2 \][/tex]
[tex]\[ q(x) = 8x^3 - 7x^2 + 4x + 2 \][/tex]
We need to subtract [tex]\( q(x) \)[/tex] from [tex]\( p(x) \)[/tex]:
[tex]\[ p(x) - q(x) = (8x^3 + 6x^2 + 5x + 2) - (8x^3 - 7x^2 + 4x + 2) \][/tex]
Expanding the subtraction:
[tex]\[ p(x) - q(x) = 8x^3 + 6x^2 + 5x + 2 - 8x^3 + 7x^2 - 4x - 2 \][/tex]
Combining like terms:
[tex]\[ p(x) - q(x) = 8x^3 - 8x^3 + 6x^2 + 7x^2 + 5x - 4x + 2 - 2 \][/tex]
[tex]\[ p(x) - q(x) = (8x^3 - 8x^3) + (6x^2 + 7x^2) + (5x - 4x) + (2 - 2) \][/tex]
[tex]\[ p(x) - q(x) = 0x^3 + 13x^2 + x + 0 \][/tex]
[tex]\[ p(x) - q(x) = 13x^2 + x \][/tex]
So, the resulting polynomial for part (c) is:
[tex]\[ 13x^2 + x \][/tex]
In summary, the differences are:
- Part (a): [tex]\(-6x^4 + 3x^3 + 6x - 6\)[/tex]
- Part (b): [tex]\(8y^3 + 3y^2 + y - 9\)[/tex]
- Part (c): [tex]\(13x^2 + x\)[/tex]
