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High School Mathematics - 2
12.10 Remainder Theorem

If f(x) is a polynomial in x and is divided by x-a and if the remainder is the value of f(x) at x = a i,e remainder = f(a).
Proof: Let p(x) be a polynomial divided by (x-a)
By division algorithm,
Dividend = (Quotient x Divisor) + Remainder
p(x) = q(x).(x-a) + R
Substitute x = a
p(a) = q(a).(a-a) + R
p(a) = R(a-a = 0, 0-q(a) = 0)
Hence remainder = p(a)

Example: Find the remainder when 2x3 + 4 is divided by (x-1)
Solution: When f(x) is divided by x-1
R = f(1)
= 2(1)3+ 4
= 12

Example: Factorise x2 +3x + 4 using remainder theorem.

  1. Find a value for the variable x for which the value of x2 - 3x + 4 is equal to 0.
  2. Let x = 1,then, f(1) = (1)2+3(1)-4 = 0, then (x-1) is a factor.
  3. Divide x2 +3x + 4 by x-1
  4. Quotient is (x+4)
Answer: Factors are (x+4) (x-1)
Directions: Solve the following.
Q 1:
Answer:

Q 2:
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Q 3:
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Q 4:
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Q 5: Factorise 2x3 + x2 - 2x - 1
Answer:

Q 6:
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Q 7:
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Q 8:
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Question 10: This question is available to subscribers only!


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