Divide each of the following polynomials by synthetic division method and also by

linear division method. Write the quotient and the remainder.

(i) $(2 m^{2} - 3 m + 10) \div (m - 5)$ (ii) $(x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5) \div (x + 2)$ (iii) $(y^{3} - 216) \div (y - 6)$

(iv) $(2 x^{4} + 3 x^{3} + 4 x - 2 x^{2}) \div (x + 3)$ (v) $(x^{4} - 3 x^{2} - 8) \div (x + 4)$ (vi) $(y^{3} - 3 y^{2} + 5 y - 1) \div (y - 1)$

Open in App

Solution

(i)
Synthetic Division:

Dividend = $2 m^{2} - 3 m + 10$

Divisor = $m - 5$

Opposite of −5 = 5

The coefficient form of the quotient is (2, 7).

∴ Quotient = 2m + 7 and Remainder = 45

Linear Method:

$2 m^{2} - 3 m + 10 = 2 m (m - 5) + 10 m - 3 m + 10 = 2 m (m - 5) + 7 (m - 5) + 35 + 10 = (m - 5) \times (2 m + 7) + 45$
(ii)
Synthetic Division:

Dividend = $x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5$

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:

$x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5 = x^{3} (x + 2) + 3 x (x + 2) - 6 x + 4 x + 5 = x^{3} (x + 2) + 3 x (x + 2) - 2 x + 5 = x^{3} (x + 2) + 3 x (x + 2) - 2 (x + 2) + 4 + 5 = (x + 2) \times (x^{3} + 3 x - 2) + 9$
(iii)
Synthetic Division:

Dividend = $y^{3} - 216 = y^{3} + 0 y^{2} + 0 y - 216$

Divisor = $y - 6$

Opposite of −6 = 6

The coefficient form of the quotient is (1, 6, 36).

∴ Quotient = y²+ 6y + 36 and Remainder = 0

Linear Method:

$y^{3} - 216 = y^{2} (y - 6) + 6 y^{2} - 216 = y^{2} (y - 6) + 6 y (y - 6) + 36 y - 216 = y^{2} (y - 6) + 6 y (y - 6) + 36 (y - 6) + 216 - 216 = y^{2} (y - 6) + 6 y (y - 6) + 36 (y - 6) = (y - 6) \times (y^{2} + 6 y + 36)$
(iv)
Synthetic Division:

Dividend = $2 x^{4} + 3 x^{3} + 4 x - 2 x^{2} = 2 x^{4} + 3 x^{3} - 2 x^{2} + 4 x + 0$

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:

$2 x^{4} + 3 x^{3} - 2 x^{2} + 4 x = 2 x^{3} (x + 3) - 6 x^{3} + 3 x^{3} - 2 x^{2} + 4 x = 2 x^{3} (x + 3) - 3 x^{2} (x + 3) + 9 x^{2} - 2 x^{2} + 4 x = 2 x^{3} (x + 3) - 3 x^{2} (x + 3) + 7 x (x + 3) - 21 x + 4 x = 2 x^{3} (x + 3) - 3 x^{2} (x + 3) + 7 x (x + 3) - 17 (x + 3) + 51 = (x + 3) \times (2 x^{3} - 3 x^{2} + 7 x - 17) + 51$
(v)
Synthetic Division:

Dividend = $x^{4} - 3 x^{2} - 8 = x^{4} + 0 x^{3} - 3 x^{2} + 0 x - 8$

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:

$x^{4} - 3 x^{2} - 8 = x^{3} (x + 4) - 4 x^{3} - 3 x^{2} - 8 = x^{3} (x + 4) - 4 x^{2} (x + 4) + 16 x^{2} - 3 x^{2} - 8 = x^{3} (x + 4) - 4 x^{2} (x + 4) + 13 x (x + 4) - 52 x - 8 = x^{3} (x + 4) - 4 x^{2} (x + 4) + 13 x (x + 4) - 52 (x + 4) + 208 - 8 = (x + 4) \times (x^{3} - 4 x^{2} + 13 x - 52) + 200$
(vi)
Synthetic Division:

Dividend = $y^{3} - 3 y^{2} + 5 y - 1$

Divisor = $y - 1$

Opposite of −1 = 1

The coefficient form of the quotient is (1, −2, 3).

∴ Quotient = y²− 2y + 3 and Remainder = 2

Linear Method:

$y^{3} - 3 y^{2} + 5 y - 1 = y^{2} (y - 1) + y^{2} - 3 y^{2} + 5 y - 1 = y^{2} (y - 1) - 2 y (y - 1) - 2 y + 5 y - 1 = y^{2} (y - 1) - 2 y (y - 1) + 3 (y - 1) + 3 - 1 = (y - 1) \times (y^{2} - 2 y + 3) + 2$

Suggest Corrections

Similar questions

(x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5) \div (x + 2)

(x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5) \div (x + 2)

(2 x^{3} - 3 x^{2} + 4 x + 2) \div (x - 1)

Divide the first polynomial by the second polynomial in each of the following, Also, write the quotient and remainder.

(i) $3 x^{2} + 4 x + 5, x - 2$

(ii) $10 x^{2} - 7 x + 8, 5 x - 3$

(iii) $5 y^{2} - 6 y^{2} + 6 y - 1, 5 y - 1$

(iv) $x^{4} - x^{3} + 5 x, x - 1$

(v) $y^{4} + y^{2}, y^{2} - 2$

p (x) = x^{4} - 3 x^{2} + 4 x + 5, g (x) = x^{2} + 1 - x

Join BYJU'S Learning Program