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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.
`(2x^4 + 3x^3 + 4x - 2x^2) ÷ (x + 3)`
Solution
Synthetic Division:
Dividend = `2x^4 + 3x^3 + 4x - 2x^2 = 2x^4 + 3x^3 - 2x^2 + 4x + 0`
Divisor = x + 3
Opposite of 3 = −3
The coefficient form of the quotient is (2, −3, 7, −17).
∴ Quotient = 2x3 − 3x2 + 7x − 17 and Remainder = 51
Linear Method:
`2x^4 + 3x^3 - 2x^2 + 4x `
`= 2x^3 (x + 3) - 6x^3 + 3x^3 -2x^2 + 4x`
`= 2x^3 (x + 3) - 3x^2 (x + 3) + 9x^2 -2x^2 +4x`
`= 2x^3 (x + 3) - 3x^2 (x + 3) +7x (x + 3) - 21x + 4x`
`= 2x^3 (x + 3) - 3x^2 (x + 3) +7x (x + 3) - 17(x + 3) + 51`
`= (x + 3) (2x^3 - 3x^2 + 7x - 17) + 51`
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