Certainly! Let's tackle the problem step-by-step, performing algebraic operations to find the simplified expression of [tex]\(A(x) - B(x) + C(x)\)[/tex].
The polynomial expressions are given as:
[tex]\[ A(x) = -2x^4 + 5x - 12 + 4x^2 \][/tex]
[tex]\[ B(x) = x^3 + 2x^5 - 3 \][/tex]
[tex]\[ C(x) = 4x^2 + 2x - 8 \][/tex]
We need to find [tex]\( A(x) - B(x) + C(x) \)[/tex].
1. Distribute and combine like terms:
[tex]\[
A(x) - B(x) + C(x) = \left(-2x^4 + 5x - 12 + 4x^2\right) - \left(x^3 + 2x^5 - 3\right) + \left(4x^2 + 2x - 8\right).
\][/tex]
2. Apply the subtraction to [tex]\( B(x) \)[/tex]:
[tex]\[
= -2x^4 + 5x - 12 + 4x^2 - x^3 - 2x^5 + 3 + 4x^2 + 2x - 8.
\][/tex]
3. Combine the like terms:
- Combine the [tex]\( x^5 \)[/tex] terms:
[tex]\[
-2x^5.
\][/tex]
- Combine the [tex]\( x^4 \)[/tex] terms:
[tex]\[
-2x^4.
\][/tex]
- Combine the [tex]\( x^3 \)[/tex] terms:
[tex]\[
-x^3.
\][/tex]
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
4x^2 + 4x^2 = 8x^2.
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
5x + 2x = 7x.
\][/tex]
- Combine the constant terms:
[tex]\[
-12 + 3 - 8 = -17.
\][/tex]
4. Write the simplified combined polynomial:
[tex]\[
A(x) - B(x) + C(x) = -2x^5 - 2x^4 - x^3 + 8x^2 + 7x - 17.
\][/tex]
Thus, the simplified expression for [tex]\( A(x) - B(x) + C(x) \)[/tex] is:
[tex]\[
-2x^5 - 2x^4 - x^3 + 8x^2 + 7x - 17.
\][/tex]
That is the final simplified polynomial as required.
