Asked by pakyo

What is the quotient of (3x4−13x3+7x2−x−9)÷(x2−4x+1)?

Answers

Answered by GPT-4o mini
To perform the polynomial division of \( (3x^4 - 13x^3 + 7x^2 - x - 9) \) by \( (x^2 - 4x + 1) \), we can use long division.

1. **Divide the leading terms:**
The leading term of the dividend \( 3x^4 \) is divided by the leading term of the divisor \( x^2 \):
\[
3x^4 \div x^2 = 3x^2
\]

2. **Multiply and subtract:**
Multiply \( 3x^2 \) by \( (x^2 - 4x + 1) \):
\[
3x^2 (x^2 - 4x + 1) = 3x^4 - 12x^3 + 3x^2
\]
Now subtract this from the original polynomial:
\[
(3x^4 - 13x^3 + 7x^2 - x - 9) - (3x^4 - 12x^3 + 3x^2) = (-13x^3 + 12x^3) + (7x^2 - 3x^2) - x - 9
\]
Simplifying this gives:
\[
-x^3 + 4x^2 - x - 9
\]

3. **Repeat for the next term:**
Now divide the leading term \( -x^3 \) by \( x^2 \):
\[
-x^3 \div x^2 = -x
\]
Multiply \( -x \) by the divisor:
\[
-x (x^2 - 4x + 1) = -x^3 + 4x^2 - x
\]
Subtract this from the previous remainder:
\[
(-x^3 + 4x^2 - x - 9) - (-x^3 + 4x^2 - x) = -9
\]

4. **Final result:**
We have the remainder \( -9 \).
Therefore, the result of the division is:
\[
3x^2 - x + \frac{-9}{x^2 - 4x + 1}
\]

So, the final quotient of the division \( (3x^4 - 13x^3 + 7x^2 - x - 9) \div (x^2 - 4x + 1) \) is:
\[
\boxed{3x^2 - x - \frac{9}{x^2 - 4x + 1}}
\]
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