# Remainder Theorem

Definition of Remainder Theorem:

Let p(x) be any polynomial of degree greater than or equal to 1 and let α be any real number. If p(x) is divided by the polynomial (x - α), then the remainder is p(α).

In other words:

If the polynomial f(x) is divided by x - α then the remainder R is given by f(x) = (x - α) q(x) + R, where q(x) is the quotient and R is a constant (because the degree of the remainder is less than the degree of the divisor x - α).

Putting x = α, f(α) = (α - α)q(α) + R or f(α) = R

When the polynomial f(x) is divided by x - α, the remainder R = f(α) =  value of f(x) when x is α.

Solved examples on Remainder Theorem:

1. Find the remainder when x$$^{3}$$ + 3x$$^{2}$$ + 3x +1 is divided by

(i) x + 1

(ii) x - $$\frac{1}{2}$$

(iii) x

(iv) x + γ

(v) 5 + 2x

Solution:

(i) Let f(x) = x$$^{3}$$ + 3x$$^{2}$$ + 3x +1, divisor is x +1

Then by the Remainder Theorem we get,

Remainder = f(-1)

= (-1)$$^{3}$$ + 3(-1)$$^{2}$$ + 3(-1) +1

= -1 + 3 - 3 + 1

= 0

(ii) Let f(x) = x$$^{3}$$ + 3x$$^{2}$$ + 3x +1, divisor is x - $$\frac{1}{2}$$

Then by the Remainder Theorem we get,

Remainder = f($$\frac{1}{2}$$)

= ($$\frac{1}{2}$$)$$^{3}$$ + 3($$\frac{1}{2}$$)$$^{2}$$ + 3($$\frac{1}{2}$$) + 1

= $$\frac{1}{8}$$ + $$\frac{3}{4}$$ + $$\frac{3}{2}$$ + 1

= $$\frac{1 + 6 + 12 + 8}{8}$$

= $$\frac{27}{8}$$

(iii) Let f(x) = x$$^{3}$$ + 3x$$^{2}$$ + 3x +1, divisor is x i.e., x - 0

Then by the Remainder Theorem we get,

Remainder = f(0)

= 0$$^{3}$$ + 3 ∙ 0$$^{2}$$ + 3 ∙  0 + 1

= 1

(iv) Let f(x) = x$$^{3}$$ + 3x$$^{2}$$ + 3x +1, divisor is x + γ

Then by the Remainder Theorem we get,

Remainder = f(-γ)

= (-γ)$$^{3}$$ + 3(-γ)$$^{2}$$ + 3(-γ) +1

= -γ$$^{3}$$ + 3γ$$^{2}$$ - 3γ +1

(v) Let f(x) = x$$^{3}$$ + 3x$$^{2}$$ + 3x +1, divisor is 5 + 2x

Then by the Remainder Theorem we get,

Remainder = f(-$$\frac{5}{2}$$)

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

= $$\frac{-125}{8}$$ + $$\frac{75}{4}$$ - $$\frac{15}{2}$$ + 1

= $$\frac{-125 + 150 -60 + 8}{8}$$

= -$$\frac{27}{8}$$

2. If 3x$$^{2}$$ - 7x + 11 is divided by x - 2 then find the remainder.

Solution:

Here p(x) = 3x$$^{2}$$ - 7x + 11, divisor is x - 2

Therefore, remainder = p(2)                    [Taking x = 2 from x - 2 = 0]

= 3(2)$$^{2}$$ - 7(2) + 11

= 12 - 14 + 11

= 9

● Factorization

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