Perform the operation(s) and simplify completely. x3−2x2+1+3x5−12x4+21x2/x2−4x (1 point) Responses −2x3−2x2+7x+1 negative 2 x cubed minus 2 x squared plus 7 x plus 1 - no response given 4x3−9x+1 4 x cubed minus 9 x plus 1 - no response given 4x3−2x2−7x+1 4 x cubed minus 2 x squared minus 7 x plus 1 - no response given 4x3−2x2+7x+1

Answer 1

To simplify the expression \(\frac{x^3 - 2x^2 + 1 + 3x^5 - 12x^4 + 21x^2}{x^2 - 4x}\), let's start with the numerator and denominator separately.

1. Combine Like Terms in the Numerator:

   The numerator is \(3x^5 - 12x^4 + x^3 + (-2x^2 + 21x^2) + 1 = 3x^5 - 12x^4 + x^3 + 19x^2 + 1\).

   So, we have: \[ \text{Numerator} = 3x^5 - 12x^4 + x^3 + 19x^2 + 1 \]

2. Factor the Denominator:

   The denominator can be factored as: \[ x^2 - 4x = x(x - 4) \]

3. Divide the Numerator by the Denominator:

   We can now perform polynomial long division to divide \(3x^5 - 12x^4 + x^3 + 19x^2 + 1\) by \(x(x - 4)\).

   a) First, divide the leading term of the numerator \(3x^5\) by the leading term of the denominator \(x^2\) to get \(3x^3\).

   b) Multiply \(3x^3\) by the entire denominator: \[ 3x^3 \cdot (x^2 - 4x) = 3x^5 - 12x^4 \]

   c) Subtract: \[ (3x^5 - 12x^4 + x^3 + 19x^2 + 1) - (3x^5 - 12x^4) = x^3 + 19x^2 + 1 \]

   d) Now, repeat the process. Divide the leading term \(x^3\) by \(x^2\) to get \(x\).

   e) Multiply: \[ x \cdot (x^2 - 4x) = x^3 - 4x^2 \]

   f) Subtract: \[ (x^3 + 19x^2 + 1) - (x^3 - 4x^2) = 23x^2 + 1 \]

   g) Now repeat again. Divide the leading term \(23x^2\) by \(x^2\) to get \(23\).

   h) Multiply: \[ 23 \cdot (x^2 - 4x) = 23x^2 - 92x \]

   i) Subtract: \[ (23x^2 + 1) - (23x^2 - 92x) = 92x + 1 \]

   Now we combine all the results: The quotient is \(3x^3 + x + 23\) and the remainder is \(92x + 1\).

   So we rewrite the original expression: \[ \frac{3x^5 - 12x^4 + x^3 + 19x^2 + 1}{x(x - 4)} = 3x^3 + x + 23 + \frac{92x + 1}{x(x - 4)} \]

Since the problem requests only the simplification of the polynomial part:

The simplified result of dividing the polynomial part is: \[ x^3 + 3x + 23 \]

Based on the usual answer choices, check for any possible polynomial varying with constants.

If just consider the main polynomial division: \[ 3x^3 + x + 23 \]

Hence the nearest representation among the options is: \[ \text{The final decision is this representation answer.} \]

No option boxed exact reflects polynomial nature here. Thus, thorough checks required through similarity criteria, checking each option yields proper selecting approach.

Final Answer:

None of the choices directly match the calculated final polynomial quotient across operations throughout, thus seek subsequent proper selections through context through refined polynomial choices may yield verification assurance on selection base.

Feel free to clarify areas that would maximize understanding across polynomial trait recap check!