To perform the polynomial division of \( (3x^4 - 13x^3 + 7x^2 - x - 9) \) by \( (x^2 - 4x + 1) \), we can use long division.
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Divide the leading terms:
The leading term of the dividend \( 3x^4 \) is divided by the leading term of the divisor \( x^2 \): \[ 3x^4 \div x^2 = 3x^2 \] -
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: \[ -x^3 + 4x^2 - x - 9 \] -
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: \[ (-x^3 + 4x^2 - x - 9) - (-x^3 + 4x^2 - x) = -9 \] -
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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