Dividing Polynomials: (x^5-x^4+6x^3-2x^2+x-8)/(x-1)
This article will demonstrate how to divide the polynomial x^5-x^4+6x^3-2x^2+x-8 by x-1 using polynomial long division.
Polynomial Long Division
Polynomial long division is a method for dividing polynomials similar to long division with numbers. Here's how it works:
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Set up the division: Write the dividend (x^5-x^4+6x^3-2x^2+x-8) inside the division symbol and the divisor (x-1) outside.
____________ x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8
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Divide the leading terms: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x). This gives us x^4. Write this term above the division symbol.
x^4 __________ x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8
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Multiply and subtract: Multiply the divisor (x-1) by the term we just wrote (x^4), which gives us x^5 - x^4. Write this result below the dividend and subtract.
x^4 __________ x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 -(x^5 - x^4) ___________ 6x^3 - 2x^2 + x - 8
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Bring down the next term: Bring down the next term of the dividend (6x^3).
x^4 __________ x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 -(x^5 - x^4) ___________ 6x^3 - 2x^2 + x - 8
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Repeat steps 2-4: Repeat the process of dividing the leading term, multiplying, and subtracting.
x^4 + 6x^2 __________ x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 -(x^5 - x^4) ___________ 6x^3 - 2x^2 + x - 8 -(6x^3 - 6x^2) ___________ 4x^2 + x - 8
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Continue until the degree of the remainder is less than the degree of the divisor: Keep repeating the process until the degree of the remainder is less than the degree of the divisor (in this case, x-1).
x^4 + 6x^2 + 4x __________ x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 -(x^5 - x^4) ___________ 6x^3 - 2x^2 + x - 8 -(6x^3 - 6x^2) ___________ 4x^2 + x - 8 -(4x^2 - 4x) ___________ 5x - 8
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Write the final result: The quotient is the expression above the division symbol: x^4 + 6x^2 + 4x. The remainder is the last expression obtained: 5x - 8.
Therefore, the result of dividing (x^5-x^4+6x^3-2x^2+x-8) by (x-1) is:
(x^5-x^4+6x^3-2x^2+x-8) / (x-1) = x^4 + 6x^2 + 4x + (5x - 8)/(x-1)
This result can be expressed as:
x^5-x^4+6x^3-2x^2+x-8 = (x-1)(x^4 + 6x^2 + 4x) + (5x - 8)
This shows how the original polynomial can be factored and written with a remainder.