Question

Divide of the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(x^4 + 2x^3 +3x^2 + 4x + 5) ÷ (x + 2)`

Answer 1

Synthetic Division:

Dividend = `x^4 + 2x^3 +3x^2 + 4x + 5`

Divisor = x + 2

Opposite of 2 = −2

The coefficient form of the quotient is (1, 0, 3, −2). 
∴ Quotient = x³ + 3x − 2 and Remainder = 9

Linear Method:

x⁴ + 2x³ + 3x² + 4x + 5 ÷ (x + 2)

x⁴ ÷ x = x³ ⇒ x³ (x + 2) = x⁴ + 2x³

(x⁴ + 2x³) − (x⁴ + 2x³) = 0

3x² ÷ x = 3x ⇒ 3x (x + 2) = 3x² + 6x

(3x² + 4x) − (3x² + 6x) = −2x

−2x ÷ x = −2 ⇒ −2(x + 2) = −2x − 4

(−2x + 5) − (−2x − 4) = 9