Sure, let's go through the subtractions step by step.
### Part (i)
Let's start with the first pair of polynomials:
[tex]\[ P_1 = 8x^2 - 4y^2 + 6xy + 5 \][/tex]
[tex]\[ P_2 = 3x^2 - 4y^2 + 5xy - 3 \][/tex]
We need to subtract [tex]\( P_2 \)[/tex] from [tex]\( P_1 \)[/tex]:
[tex]\[ P_1 - P_2 = (8x^2 - 4y^2 + 6xy + 5) - (3x^2 - 4y^2 + 5xy - 3) \][/tex]
Step-by-step process:
1. Subtract the [tex]\(x^2\)[/tex] terms:
[tex]\[ 8x^2 - 3x^2 = 5x^2 \][/tex]
2. Subtract the [tex]\(y^2\)[/tex] terms (note that [tex]\(-4y^2 - (-4y^2) = -4y^2 + 4y^2 = 0\)[/tex]):
[tex]\[ -4y^2 + 4y^2 = 0 \][/tex]
3. Subtract the [tex]\(xy\)[/tex] terms:
[tex]\[ 6xy - 5xy = xy \][/tex]
4. Subtract the constant terms:
[tex]\[ 5 - (-3) = 5 + 3 = 8 \][/tex]
So, combining all these results, we get:
[tex]\[ P_1 - P_2 = 5x^2 + xy + 8 \][/tex]
### Part (ii)
Now, let's handle the second pair of polynomials:
[tex]\[ Q_1 = 3x^2y - 4xy^2 + 5xy \][/tex]
[tex]\[ Q_2 = x^2y - 3xy^2 + 4xy \][/tex]
We need to subtract [tex]\( Q_2 \)[/tex] from [tex]\( Q_1 \)[/tex]:
[tex]\[ Q_1 - Q_2 = (3x^2y - 4xy^2 + 5xy) - (x^2y - 3xy^2 + 4xy) \][/tex]
Step-by-step process:
1. Subtract the [tex]\(x^2y\)[/tex] terms:
[tex]\[ 3x^2y - x^2y = 2x^2y \][/tex]
2. Subtract the [tex]\(xy^2\)[/tex] terms:
[tex]\[ -4xy^2 - (-3xy^2) = -4xy^2 + 3xy^2 = -xy^2 \][/tex]
3. Subtract the [tex]\(xy\)[/tex] terms:
[tex]\[ 5xy - 4xy = xy \][/tex]
So, combining all these results, we get:
[tex]\[ Q_1 - Q_2 = 2x^2y - xy^2 + xy \][/tex]
### Summary
- For part (i), the result is:
[tex]\[ 5x^2 + xy + 8 \][/tex]
- For part (ii), the result is:
[tex]\[ 2x^2y - xy^2 + xy \][/tex]
These are the simplified polynomials after performing the subtractions.
