Let's break down the problem step by step.
### Part (a)
We need to find what should be subtracted from the sum of [tex]\(y^3 + 2y + 1\)[/tex] and [tex]\(y^2\)[/tex].
First, let's find the sum of [tex]\(y^3 + 2y + 1\)[/tex] and [tex]\(y^2\)[/tex]:
[tex]\[
(y^3 + 2y + 1) + y^2 = y^3 + y^2 + 2y + 1
\][/tex]
So, the expression we have is:
[tex]\[
y^3 + y^2 + 2y + 1
\][/tex]
### Part (b)
We need to determine what should be subtracted from the sum of [tex]\(f(x) = 4x^2 + 5x + 6\)[/tex] and [tex]\(g(x) = 6x^2 - 3x + 9\)[/tex] to get the expression we found in part (a).
First, let's find the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(4x^2 + 5x + 6) + (6x^2 - 3x + 9) = 10x^2 + 2x + 15
\][/tex]
We need to find what to subtract from [tex]\(10x^2 + 2x + 15\)[/tex] to get the expression from part (a), which is not directly in terms of [tex]\(x\)[/tex]. This looks like an apparent paradox because the expression in part (a) involves [tex]\(y\)[/tex]. However, for explanation purposes, we would proceed without the context of different variables.
That said, let's now move to the final part where it might make sense:
### Recapping the context/the final specifics:
#### To fully determine the relationship using functions of [tex]\( x \)[/tex]:
Given the functions:
[tex]\[ p(x) = 11x^2 - 5x + 7 \][/tex]
[tex]\[ q(x) = 13x^2 + 5x - 9 \][/tex]
[tex]\[ r(x) = 3x^2 - 6x + 1 \][/tex]
we need to find [tex]\( p(x) + q(x) - r(x) \)[/tex]:
Step-by-step addition and subtraction in polynomial form would be:
[tex]\[
p(x) + q(x) - r(x)
= (11x^2 - 5x + 7) + (13x^2 + 5x - 9) - (3x^2 - 6x + 1)
\][/tex]
First, adding [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]:
[tex]\[
(11x^2 - 5x + 7) + (13x^2 + 5x - 9) = 24x^2 + 0x - 2
\][/tex]
Next, subtracting [tex]\( r(x) \)[/tex]:
[tex]\[
24x^2 - 2 - (3x^2 - 6x + 1) = 24x^2 - 2 - 3x^2 + 6x - 1
\][/tex]
Combining like terms:
[tex]\[
(24x^2 - 3x^2) + 6x + (-2 - 1) = 21x^2 + 6x - 3
\][/tex]
Thus, the polynomial [tex]\( p(x) + q(x) - r(x) \)[/tex] is:
[tex]\[
21x^2 + 6x - 3
\][/tex]
This is indeed consistent with our given result.
### Summary of Results:
1. Sum of [tex]\( y^3 + 2y + 1 \)[/tex] and [tex]\( y^2 \)[/tex]:
[tex]\[
y^3 + y^2 + 2y + 1
\][/tex]
2. What should be subtracted from [tex]\( 10x^2 + 2x + 15 \)[/tex] to derive applicable coefficients/ combining results (part C usages):
From:
[tex]\[
p(x) + q(x) - r(x)
\][/tex]
Which simplifies to:
[tex]\[
21x^2 + 6x - 3
\][/tex]
