%2F(x%5E2%2B8).jpeg "(6x^4+x^3-9x+13)/(x^2+8)")
Polynomial Long Division: A Step-by-Step Guide
In this article, we will delve into the process of performing polynomial long division on the expression (6x⁴ + x³ - 9x + 13)/(x² + 8).
Understanding Polynomial Long Division
Polynomial long division is a method for dividing polynomials, similar to long division with numbers. It allows us to find the quotient and remainder of dividing a polynomial (the dividend) by another polynomial (the divisor).
Steps for Polynomial Long Division
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Set up the division: Write the dividend (6x⁴ + x³ - 9x + 13) inside the division symbol and the divisor (x² + 8) outside.
___________ x² + 8 | 6x⁴ + x³ - 9x + 13 -
Focus on the leading terms: Divide the leading term of the dividend (6x⁴) by the leading term of the divisor (x²). This gives us 6x².
6x² ______ x² + 8 | 6x⁴ + x³ - 9x + 13 -
Multiply the quotient by the divisor: Multiply 6x² by (x² + 8) to get 6x⁴ + 48x².
6x² ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² -
Subtract: Subtract the result from the dividend, remembering to change the signs of the terms being subtracted.
6x² ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x -
Bring down the next term: Bring down the next term of the dividend (-9x).
6x² ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x -
Repeat steps 2-5: Now, focus on the new leading term (x³). Divide x³ by x² to get x.
6x² + x ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x x³ + 8xMultiply x by (x² + 8) and subtract:
6x² + x ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x x³ + 8x ------- -56x² - 17x -
Continue until the degree of the remainder is less than the degree of the divisor: Bring down the next term (13).
6x² + x ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x x³ + 8x ------- -56x² - 17x + 13Divide -56x² by x² to get -56:
6x² + x - 56 ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x x³ + 8x ------- -56x² - 17x + 13 -56x² - 448Multiply -56 by (x² + 8) and subtract:
6x² + x - 56 ______ x² + 8 | 6x⁴ + x³ - 9x + 13 6x⁴ + 48x² --------- x³ - 48x² - 9x x³ + 8x ------- -56x² - 17x + 13 -56x² - 448 -------- -17x + 461 -
The final remainder: We have reached a point where the degree of the remainder (-17x + 461) is less than the degree of the divisor (x² + 8).
The Result
Therefore, the result of the polynomial long division is:
(6x⁴ + x³ - 9x + 13)/(x² + 8) = 6x² + x - 56 + (-17x + 461)/(x² + 8)
This can also be expressed as:
6x⁴ + x³ - 9x + 13 = (6x² + x - 56)(x² + 8) + (-17x + 461)