# Remainder Theorem

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**Remainder theorem : Let P(x) be any polynomial of degree greater than or equal to one and let ‘a’ be any real number. If P(x) is divided by the linear polynomial x – a, then the remainder is P(a). If P(a) = 0 then x - a is the factor of the given polynomial.**

**Examples :**

1) Find the remainder when x

^{4}- x

^{3}- 3x

^{2}-2x +1 is divided by x – 1 .

**Solution**:

P(x) = x

^{4}- x

^{3}- 3x

^{2}-2x + 1 and

x-1 = 0

x = 1

So put x = 1 in P(x)

P(1) = (1)

^{4}- (1)

^{3}- 3(1)

^{2}-2(1) +1

= 1 -1 – 3 -2 +1

= - 5 + 1

= - 4

So, by remainder theorem, - 4 is the remainder when x

^{4}-x

^{3}-3x

^{2}-2x +1 is divided by x – 1.

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2) Find the remainder when p(x) = 4x

^{3}- 12x

^{2}+ 14x -3 is divided by

g(x) = x -1/2.

**Solution :**

By remainder theorem

g(x) = x - 1/2 ⇒ x = 1/2

Put x = 1/2 in the given polynomial

p(1/2) = 4(1/2)

^{3}- 12(1/2)

^{2}+ 14(1/2) -3

= 4/8 - 12/4 + 7 - 3

= 1/2 - 3 + 4

= 1/2 + 1

p(1/2) = 3/2.

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3) Let R1 and R2 are the remainders of when the polynomials

x

^{3}+ 2x

^{2}- 5ax - 7 and x

^{3}+ ax

^{2}-12x + 6 are divided by x + 1 and x - 2 respectively. If 2R1 + R2 = 6, find the value of a.

**Solution :**

Let p(x) = x

^{3}+ 2x

^{2}- 5ax - 7 and q(x) = x

^{3}+ ax

^{2}-12x + 6

⇒ R1 = p(-1)

⇒ R1 = (-1)

^{3}+ 2(-1)

^{2}- 5a(-1) -7

⇒ R1 = -1 + 2 + 5a - 7

⇒ R1 = 5a - 6

And,

R2 = Remainder when q(x) is divided by x - 2

R2 = q(2)

⇒ R2 = (2)

^{3}+ a(2)

^{2}-12(2) + 6

⇒ R2 = 8 + 4a - 24 + 6

⇒ R2 = 4a - 10

As 2R1 + R2 = 6

Put the values of R1 and R2 in the above equation

2(5a - 6) + 4a - 10 = 6

10a - 12 + 4a - 10 = 6

14a - 22 = 6

14a = 28

a = 28/14

a = 2

**Polynomial**

• Degree of the Polynomial

• Zeros of Polynomial

• Remainder Theorem

• Find remainder by Synthetic Division

• Rational root test in Polynomial

• Solved Examples on Polynomial

• Degree of the Polynomial

• Zeros of Polynomial

• Remainder Theorem

• Find remainder by Synthetic Division

• Rational root test in Polynomial

• Solved Examples on Polynomial

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