%2F(x%5E2-7x%2B1).jpeg "(x^3+2x^2-4x+8)/(x^2-7x+1)")
Polynomial Division: (x^3+2x^2-4x+8)/(x^2-7x+1)
This article will guide you through the process of dividing the polynomial (x^3+2x^2-4x+8) by (x^2-7x+1) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is a method used to divide polynomials, similar to the long division process we use with numbers. It involves a series of steps to find the quotient and remainder of the division.
Steps for Polynomial Long Division:
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Set up the division: Write the dividend (x^3+2x^2-4x+8) inside the division symbol and the divisor (x^2-7x+1) outside.
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Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x^2). This gives us x. Write this term above the division symbol, aligned with the x^3 term.
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Multiply the quotient: Multiply the quotient (x) by the divisor (x^2-7x+1). This gives us x^3 - 7x^2 + x. Write this below the dividend, aligning terms with matching degrees.
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Subtract: Subtract the expression you just wrote from the dividend. This gives us 9x^2 - 5x + 8.
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Bring down the next term: Bring down the next term of the dividend (-4x), giving us 9x^2 - 5x + 8.
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Repeat steps 2-5: Now repeat steps 2-5 using 9x^2 - 5x + 8 as your new dividend. Divide the leading term (9x^2) by the leading term of the divisor (x^2), which gives us 9. Write 9 next to the x in the quotient. Multiply 9 by the divisor (x^2 - 7x + 1) and write the result below the new dividend. Subtract to get 62x - 1.
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Stop when the degree of the remainder is less than the degree of the divisor: We stop here because the degree of the remainder (62x - 1) is 1, which is less than the degree of the divisor (x^2-7x+1).
The Result:
The result of the polynomial division is:
(x^3+2x^2-4x+8)/(x^2-7x+1) = x + 9 + (62x-1)/(x^2-7x+1)
This means that the quotient is x + 9 and the remainder is 62x - 1.
Conclusion
Polynomial long division can be used to divide any two polynomials. The process involves repeatedly dividing the leading term of the dividend by the leading term of the divisor, multiplying, and subtracting. The division is complete when the degree of the remainder is less than the degree of the divisor.