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Question
Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.
`(x^4 - 3x^2 - 8) ÷ (x + 4)`
Solution
Synthetic Division:
Dividend = `x^4 - 3x^2 - 8 = x^4 +0x^3 - 3x^2 + 0x - 8`
Divisor = x + 4
Opposite of 4 = −4
The coefficient form of the quotient is (1, −4, 13, −52).
∴ Quotient = x3 − 4x2 + 13x − 52 and Remainder = 200
Linear Method:
`x^4 - 3x^2 - 8`
`= x^3 (x + 4) - 4x^3 - 3x^2 - 8`
`= x^3 (x + 4) - 4x^2 (x +4) + 16x^2 -3x^2 - 8`
`= x^3 (x + 4) - 4x^2 (x + 4) + 13x (x +4) - 52x - 8`
`= x^3 (x + 4) - 4x^2 (x + 4) + 13x (x +4) - 52(x + 4) + 208 - 8`
`= (x + 4) (x^3 - 4x^2 + 13 x - 52) + 200`
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