%2F(x%5E2%2B1)-long-division.jpeg "(x^3-9)/(x^2+1) Long Division")
Long Division of (x^3 - 9) / (x^2 + 1)
This article will guide you through the process of performing long division with the expressions (x^3 - 9) and (x^2 + 1).
Understanding Long Division with Polynomials
Long division with polynomials is very similar to long division with numbers. The goal is to find a quotient and remainder that satisfy the equation:
Dividend = (Divisor * Quotient) + Remainder
In our case:
- Dividend: x^3 - 9
- Divisor: x^2 + 1
Steps for Long Division
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Set up the division: Write the dividend (x^3 - 9) inside the division symbol and the divisor (x^2 + 1) outside. Remember to include placeholders for missing terms (e.g., 0x^2 and 0x in the dividend).
________ x^2+1 | x^3 + 0x^2 + 0x - 9 -
Divide the leading terms: Focus on the leading terms of the dividend (x^3) and the divisor (x^2). What do we need to multiply x^2 by to get x^3? The answer is x.
x x^2+1 | x^3 + 0x^2 + 0x - 9 -
Multiply the divisor by the quotient term: Multiply the divisor (x^2 + 1) by the quotient term (x).
x x^2+1 | x^3 + 0x^2 + 0x - 9 x^3 + 0x^2 + x -
Subtract: Subtract the result from the dividend.
x x^2+1 | x^3 + 0x^2 + 0x - 9 x^3 + 0x^2 + x ------------- -x - 9 -
Bring down the next term: Bring down the next term from the dividend (-9).
x x^2+1 | x^3 + 0x^2 + 0x - 9 x^3 + 0x^2 + x ------------- -x - 9 -
Repeat steps 2-5: Now, focus on the new leading term (-x) and the divisor (x^2 + 1). Since x^2 doesn't divide into -x, our quotient term is 0.
x + 0 x^2+1 | x^3 + 0x^2 + 0x - 9 x^3 + 0x^2 + x ------------- -x - 9 0x^2 -x - 9 -
Find the remainder: Since we can't divide further, our remainder is -x - 9.
x + 0 x^2+1 | x^3 + 0x^2 + 0x - 9 x^3 + 0x^2 + x ------------- -x - 9 0x^2 -x - 9 ------------- -x - 9
Conclusion
Therefore, the long division of (x^3 - 9) by (x^2 + 1) results in:
Quotient: x Remainder: -x - 9
We can express this as:
(x^3 - 9) / (x^2 + 1) = x + (-x - 9) / (x^2 + 1)