To solve [tex]\( [B(x)-c(x)] \cdot 2c(x) \)[/tex] step-by-step, let's first break down [tex]\( B(x) \)[/tex] and [tex]\( c(x) \)[/tex] individually.
Firstly:
[tex]\[ B(x) = x^3 - 2x^5 \][/tex]
Then:
[tex]\[ c(x) = -4x^2 + 2x - 8 \][/tex]
To address the expression [tex]\( [B(x) - c(x)] \)[/tex], we need to plug in the values for [tex]\( B(x) \)[/tex] and [tex]\( c(x) \)[/tex] and simplify step-by-step:
[tex]\[
[B(x) - c(x)]
= [x^3 - 2x^5 - (-4x^2 + 2x - 8)]
= [x^3 - 2x^5 + 4x^2 - 2x + 8]
\][/tex]
Now, considering the full expression provided:
[tex]\[
\left[x^3 - 2x^5 - 3\left(-4x^2 + 2x - 8\right)\right] - 4x^2 + 2x - 8
\][/tex]
Let's break down the term:
[tex]\[
- 3(-4x^2 + 2x - 8) = 3 \times 4x^2 - 3 \times 2x + 3 \times 8 = 12x^2 - 6x + 24
\][/tex]
Substituting this back into the expression yields:
[tex]\[
\left[x^3 - 2x^5 + 12x^2 - 6x + 24 \right] - 4x^2 + 2x - 8
\][/tex]
Now, simplify by combining like terms:
- Combine [tex]\(4x^2\)[/tex] terms: [tex]\(12x^2 - 4x^2 = 8x^2\)[/tex]
- Combine constant terms: [tex]\(24 - 8 = 16\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(-6x + 2x = -4x\)[/tex]
Thus, the simplified form is:
[tex]\[
-2x^5 + x^3 + 8x^2 - 4x + 16
\][/tex]
This is the expanded and simplified form of the given expression:
[tex]\[
-2x^5 + x^3 + 8x^2 - 4x + 16
\][/tex]
