Find the reminder of Polynomial.

1. Find the reminder P(x) = x^3 + 5 x^2 + 5x + 1

a). x + 1

b) x – 1

c). x – 1/3

d). x + ¼

e). x

f). 3x + 6

Solution.

a). P(x) = x^3 + 5 x^2 + 5x + 1

Put x + 1 = 0 => x = -1

P(-1) = (-1)^3 + 5(-1)^2 + 5(-1) + 1

= -1 + 5 – 5 + 1 = 0

So the reminder is zero.

b). P(x) = x^3 + 5 x^2 + 5x + 1

Put x - 1 = 0 => x = 1

P(1) = (1)^3 + 5(1)^2 + 5(1) + 1

= 1 + 5 + 5 + 1 = 12

So the reminder is 12.

c). P(x) = x^3 + 5 x^2 + 5x + 1

Put x – 1/3 = 0 => x = 1/3

P(1/3) = (1/3)^3 + 5(1/3)^2 + 5(1/3) + 1

= 1/27 + 5/9 + 5/3 + 1 = ( 1 + 15 + 45 + 27 ) / 27 = 88/27

So the reminder is 88/27.

d). P(x) = x^3 + 5 x^2 + 5x + 1

Put x + 1/4 = 0 => x = -1/4

P(-1/4) = (-1/4)^3 + 5(-1/4)^2 + 5(-1/4) + 1

= -1/64 + 5/16 - 5/4 + 1 = ( -1 + 20 - 80 + 64 ) / 64 = 3/64

So the reminder is 3/64.

e). P(x) = x^3 + 5 x^2 + 5x + 1

Put x = 0

P(0) = (0)^3 + 5(0)^2 + 5(0) + 1

= 0 + 0 + 0 + 1 = 1

So the reminder is 1.

f). P(x) = x^3 + 5 x^2 + 5x + 1

Put 3x + 6 = 0 => 3x = -6

X = -6/3 => x = -2

P(-2) = (-2)^3 + 5(-2)^2 + 5(-2) + 1

= -8 + 5 * 4 – 5 * 2 + 1 = -8 + 20 – 10 + 1 = 3

So the reminder is 3

maths-ncert-solution

2. Find the reminder of P(x) = 3x^3 – a x^2 + 2 x – 2 a when divided by x – a

Sol. P(x) = 3x^3 – a x^2 + 2 x – 2 a

Put x – a = 0 => x = a

P(a) = 3 a^3 – a * a^2 + 2 * a -2 a

= 3 a^3 – a^3 + 2a -2a

= 2 a^3

So the reminder is 2 a^3

3. Find the reminder of P(x) = 27 x^3 – 9 x^2 + 3 x – 1 when divided by x – 1/3

Sol. P(x) = 27 x^3 – 9 x^2 + 3 x – 1

Put x – 1/3 = 0 => x = 1/3

P(1/3) = 27 * (1/3)^3 – 9 * (1/3)^2 + 3 * (1/3) - 1

= 1 – 1 + 1 – 1 = 0

So the reminder is 0.

Milan Tomic

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