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### High School Mathematics - 212.10 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. Find a value for the variable x for which the value of x2 - 3x + 4 is equal to 0. Let x = 1,then, f(1) = (1)2+3(1)-4 = 0, then (x-1) is a factor. Divide x2 +3x + 4 by x-1 Quotient is (x+4) Answer: Factors are (x+4) (x-1) Directions: Solve the following.
 Q 1: Factorise 2x3 + x2 - 2x - 1Answer: Q 2: Answer: Q 3: Answer: Q 4: Answer: Q 5: Factorise completely 2x3 - 7x2 - 3x + 18 completelyAnswer: Q 6: Answer: Q 7: Answer: Q 8: Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!