**Chapter 3 Polynomials Practice Set 3.3**

Chapter 3 Polynomials Practice Set 3.3

**Question 1.Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder.i. (2m**

^{2}– 3m + 10) ÷ (m – 5)ii. (x

^{4}+ 2x

^{3}+ 3x

^{2}+ 4x + 5) ÷ (x + 2)iii. (y

^{3}– 216) ÷ (y – 6)iv. (2x

^{4}+ 3x

^{3}+ 4x – 2x

^{2}) ÷ (x + 3)v. (x

^{4}– 3x

^{2}– 8) ÷ (x + 4)vi. (y

^{3}– 3y

^{2}+ 5y – 1) ÷ (y – 1)Solution:

i. Synthetic division:

(2m

^{2}– 3m + 10) ÷ (m – 5)

Dividend = 2m² – 3m + 10

∴ Coefficient form of dividend = (2, -3, 10)

Divisor = m – 5

∴ Opposite of -5 is 5.

Coefficient form of quotient = (2, 7)

∴ Quotient = 2m + 7,

Remainder = 45

Linear division method:

2m

^{2}– 3m + 10

To get the term 2m

^{2}, multiply (m – 5) by 2m and add 10m,

= 2m(m – 5) + 10m- 3m + 10

= 2m(m – 5) + 7m + 10

To get the term 7m, multiply (m – 5) by 7 and add 35

= 2m(m – 5) + 7(m- 5) + 35+ 10

= (m – 5) (2m + 7) + 45

∴ Quotient = 2m + 7,

Remainder = 45

ii. Synthetic division:

(x^{4} + 2x^{3} + 3x^{2} + 4x + 5) ÷ (x + 2)

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

∴ Coefficient form of dividend = (1, 2, 3, 4, 5)

Divisor = x + 2

∴ Opposite of + 2 is -2.

Coefficient form of quotient = (1, 0, 3, -2)

∴ Quotient = x^{3} + 3x – 2,

Remainder = 9

Linear division method:

x^{4} + 2x^{3} + 3x^{2} + 4x + 5

To get the term x^{4}, multiply (x + 2) by x^{3} and subtract 2x^{3},

= x^{3}(x + 2) – 2x^{3} + 2x^{3} + 3x^{2} + 4x + 5

= x^{3}(x + 2) + 3x^{2} + 4x + 5

To get the term 3x^{2}, multiply (x + 2) by 3x and subtract 6x,

= x^{3}(x + 2) + 3x(x + 2) – 6x + 4x + 5

= x^{3}(x + 2) + 3x(x + 2) – 2x + 5

To get the term -2x, multiply (x + 2) by -2 and add 4,

= x^{3}(x + 2) + 3x(x + 2) – 2(x + 2) + 4 + 5

= (x + 2) (x3 + 3x – 2) + 9

∴ Quotient = x^{3} + 3x – 2,

Remainder – 9

iii. Synthetic division:

(y^{3} – 216) ÷ (y – 6)

Dividend = y^{3} – 216

∴ Index form = y^{3} + 0y^{3} + 0y – 216

∴ Coefficient form of dividend = (1, 0, 0, -216)

Divisor = y – 6

∴ Opposite of – 6 is 6.

Coefficient form of quotient = (1, 6, 36)

∴ Quotient = y^{2} + 6y + 36,

Remainder = 0

Linear division method:

y^{3} – 216

To get the term y^{3}, multiply (y – 6) by y^{2} and add 6y^{2},

= y^{2}(y – 6) + 6y^{2} – 216

= y^{2}(y – 6) + 6ysup>2 – 216

To get the, term 6 y^{2} multiply (y – 6) by 6y and add 36y,

= y^{2}(y – 6) + 6y(y – 6) + 36y – 216

= y^{2}(y – 6) + 6y(y – 6) + 36y – 216

To get the term 36y, multiply (y- 6) by 36 and add 216,

= y^{2} (y – 6) + 6y(y – 6) + 36(y – 6) + 216 – 216

= (y – 6) (y^{2} + 6y + 36) + 0

Quotient = y^{2} + 6y + 36

Remainder = 0

iv. Synthetic division:

(2x^{4} + 3x^{3} + 4x – 2x^{2}) ÷ (x + 3)

Dividend = 2x^{4} + 3x^{3} + 4x – 2x^{2}

∴ Index form = 2x^{4} + 3x^{3} – 2x^{2} + 4x + 0

∴ Coefficient form of the dividend = (2,3, -2,4,0)

Divisor = x + 3

∴ Opposite of + 3 is -3

Coefficient form of quotient = (2, -3, 7, -17)

∴ Quotient = 2x^{3} – 3x^{2} + 7x – 17,

Remainder = 51

Linear division method:

2x^{4} + 3x^{3} + 4x – 2x^{2} = 2x^{2} + 3x^{3} – 2x^{2} + 4x

To get the term 2x^{4}, multiply (x + 3) by 2x^{3} and subtract 6x^{3},

= 2x^{3}(x + 31 – 6x^{3} + 3x^{3} – 2x^{2} + 4x

= 2x^{3}(x + 3) – 3x^{3} – 2x^{2} + 4x

To get the term – 3x^{3}, multiply (x + 3) by -3x^{2} and add 9x^{2},

= 2x^{3}(x + 3) – 3x^{2}(x + 3) + 9x^{2} – 2x^{2} + 4x

= 2x^{3}(x + 3) – 3x^{2}(x + 3) + 7x^{2} + 4x

To get the term 7x^{2}, multiply (x + 3) by 7x and subtract 21x,

= 2x^{3}(x + 3) – 3x^{2}(x + 3) + 7x(x + 3) – 21x + 4x

= 2x^{3}(x + 3) – 3x^{2}(x + 3) + 7x(x + 3) – 17x

To get the term -17x, multiply (x + 3) by -17 and add 51,

= 2x^{3}(x + 3) – 3x^{2}(x + 3) + 7x(x+3) – 17(x + 3) + 51

= (x + 3) (2x^{3} – 3x^{2} + 7x- 17) + 51

∴ Quotient = 2x^{3} – 3x^{2} + 7x – 17,

Remainder = 51

v. Synthetic division:

(x^{4} – 3x^{2} – 8) + (x + 4)

Dividend = x^{4} – 3x^{2} – 8

∴ Index form = x^{4} + 0x^{3} – 3x^{2} + 0x – 8

∴ Coefficient form of the dividend = (1,0, -3,0, -8)

Divisor = x + 4

∴ Opposite of + 4 is -4

∴ Coefficient form of quotient = (1, -4, 13, -52)

∴ Quotient = x^{3} – 4x^{2} + 13x – 52,

Remainder = 200

Linear division method:

x^{4} – 3x^{2} – 8

To get the term x^{4}, multiply (x + 4) by x^{3} and subtract 4x^{3},

= x^{3}(x + 4) – 4x^{3} – 3x^{2} – 8

= x^{3}(x + 4) – 4x^{3} – 3x^{2} – 8

To get the term – 4x^{3}, multiply (x + 4) by -4x^{2} and add 16x^{2},

= x^{3}(x + 4) – 4x^{2} (x + 4) + 16x^{2} – 3x^{2} – 8

= x^{3}(x + 4) – 4x^{2} (x + 4) + 13x^{2} – 8

To get the term 13x^{2}, multiply (x + 4) by 13x and subtract 52x,

= x^{3}(x + 4) – 4x^{2}(x + 4) + 13x(x + 4) – 52x – 8

= x^{3}(x + 4) – 4x^{2}(x + 4) + 13x(x + 4) – 52x – 8

To get the term -52x, multiply (x + 4) by – 52 and add 208,

= x^{3}(x + 4) – 4x^{2}(x + 4) + 13x(x + 4) – 52(x + 4) + 208 – 8

= (x + 4) (x^{3} – 4x^{2} + 13x – 52) + 200

∴ Quotient = x^{3} – 4x^{2} + 13x – 52,

Remainder 200

vi. Synthetic division:

(y^{3} – 3y^{2} + 5y – 1) ÷ (y – 1)

Dividend = y^{3} – 3y^{2} + 5y – 1

Coefficient form of the dividend = (1, -3, 5, -1)

Divisor = y – 1

∴Opposite of -1 is 1.

∴ Coefficient form of quotient = (1, -2, 3)

∴ Quotient = y^{2} – 2y + 3,

Remainder = 2

Linear division method:

y^{3} -3y^{2} + 5y – 1

To get the term y^{3} , multiply (y – 1) by y^{2} and add y^{2}

= y^{2} (y – 1) + y^{2} – 3y^{2} + 5y – 1

= y^{2} (y – 1) – 2y^{2} + 5y – 1

To get the term -2y^{2}, multiply (y – 1) by -2y and subtract 2y,

= y^{2} (y – 1) – 2y(y – 1) – 2y + 5y – 1

= y^{2} (y – 1) – 2y(y – 1) + 3y – 1

To get the term 3y, multiply (y – 1) by 3 and add 3,

= y^{2} (y – 1) – 2y(y – 1) + 3(y- 1) + 3 – 1

= (y – 1)(y^{2} – 2y + 3) + 2

∴ Quotient = y^{2} – 2y + 3,

Remainder = 2.