To simplify the expression \( 8x^3 - 12x^2 + 18x - 27 - 2x - 3 - (x^2 + 4x) \), we will first distribute the negative sign through the parentheses.
The original expression is:
\[ 8x^3 - 12x^2 + 18x - 27 - 2x - 3 - (x^2 + 4x) \]
Distributing the negative sign gives:
\[ 8x^3 - 12x^2 + 18x - 27 - 2x - 3 - x^2 - 4x \]
Now, combine like terms. Start with the \(x^3\) terms, then \(x^2\) terms, followed by \(x\) terms, and finally the constant terms:
- Cubic term: \(8x^3\) (there's only one)
- Quadratic terms: \(-12x^2 - x^2 = -13x^2\)
- Linear terms: \(18x - 2x - 4x = 12x\)
- Constant terms: \(-27 - 3 = -30\)
Putting it all together, the expression simplifies to:
\[ 8x^3 - 13x^2 + 12x - 30 \]
Now, we can identify \(A\), \(B\), and \(C\) in the standard polynomial form \(Ax^3 + Bx^2 + Cx + D\):
- \(A = 8\)
- \(B = -13\)
- \(C = 12\)
- The constant term \(D = -30\)
So, the values of \(A\), \(B\), and \(C\) are:
\[ A = 8, \quad B = -13, \quad C = 12 \]
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