%2F(x%5E2%2B3).jpeg "(x^5-4x^4+4x^3-13x^2+3x-1)/(x^2+3)")
Polynomial Long Division: (x^5-4x^4+4x^3-13x^2+3x-1)/(x^2+3)
In this article, we will perform polynomial long division to find the quotient and remainder of the division:
(x^5-4x^4+4x^3-13x^2+3x-1) / (x^2+3)
Steps of Polynomial Long Division
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Set up the division: Write the dividend (x^5-4x^4+4x^3-13x^2+3x-1) inside the division symbol and the divisor (x^2+3) outside.
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Focus on the leading terms: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x^2). This gives us x^3. Write this term above the division symbol, aligned with the x^5 term.
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Multiply the quotient by the divisor: Multiply x^3 by (x^2+3) to get x^5 + 3x^3. Write this result below the dividend, aligning terms with corresponding powers.
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Subtract: Subtract the result from the dividend. Notice that the x^5 terms cancel out.
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Bring down the next term: Bring down the next term of the dividend (-4x^4).
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Repeat steps 2-5: Now, we focus on the new leading term (-4x^4) and divide it by the leading term of the divisor (x^2). This gives us -4x^2. Write this term above the division symbol, aligned with the x^4 term. Multiply -4x^2 by (x^2+3) to get -4x^4 - 12x^2, and subtract it from the current expression.
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Continue the process: Repeat steps 2-5 until the degree of the remaining polynomial is less than the degree of the divisor.
Performing the Division
Following these steps, we can perform the polynomial long division as shown below:
x^3 - 4x^2 + 7x - 26
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x^2+3 | x^5 - 4x^4 + 4x^3 - 13x^2 + 3x - 1
-(x^5 + 3x^3)
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-4x^4 + x^3 - 13x^2
-(-4x^4 - 12x^2)
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x^3 - x^2 + 3x
-(x^3 + 3x)
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-x^2 + 3x - 1
-(-x^2 - 3)
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3x + 2
Result
Therefore, the result of the polynomial long division is:
(x^5-4x^4+4x^3-13x^2+3x-1) / (x^2+3) = x^3 - 4x^2 + 7x - 26 + (3x + 2)/(x^2 + 3)
The quotient is x^3 - 4x^2 + 7x - 26 and the remainder is 3x + 2.