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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