Certainly! Let's go through the step-by-step process to subtract the given polynomials:
Given polynomials:
[tex]\[ P(y) = 7y^3 + 4y^2 - y - 9 \][/tex]
[tex]\[ Q(y) = y^2 - 9y + 4 \][/tex]
We need to subtract [tex]\( Q(y) \)[/tex] from [tex]\( P(y) \)[/tex].
[tex]\[
(P(y) - Q(y)) = (7y^3 + 4y^2 - y - 9) - (y^2 - 9y + 4)
\][/tex]
To perform the subtraction, distribute the negative sign through [tex]\( Q(y) \)[/tex]:
[tex]\[
= 7y^3 + 4y^2 - y - 9 - y^2 + 9y - 4
\][/tex]
Now, let's combine like terms:
1. [tex]\( y^3 \)[/tex] term:
- There is only one [tex]\( y^3 \)[/tex] term:
[tex]\[ 7y^3 \][/tex]
2. [tex]\( y^2 \)[/tex] terms:
- Combine [tex]\( 4y^2 \)[/tex] and [tex]\( -y^2 \)[/tex]:
[tex]\[ 4y^2 - y^2 = 3y^2 \][/tex]
3. [tex]\( y \)[/tex] terms:
- Combine [tex]\( -y \)[/tex] and [tex]\( 9y \)[/tex]:
[tex]\[ -y + 9y = 8y \][/tex]
4. Constant terms:
- Combine [tex]\( -9 \)[/tex] and [tex]\( -4 \)[/tex]:
[tex]\[ -9 - 4 = -13 \][/tex]
Putting all these together, the expression simplifies to:
[tex]\[
7y^3 + 3y^2 + 8y - 13
\][/tex]
Thus, the difference is [tex]\( 7y^3 + 3y^2 + 8y - 13 \)[/tex].
