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| Remainder Theorem |
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| Recall the Remainder theorem and the Factor theorem. |
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| When f(x) is divided by (x-a) the remainder is (x-a) and if the remainder f(a) = 0 then x - a is a factor of the expression f (x). |
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| If f(x) is a polynomial in x and is divided by x-a; the remainder is the value of f(x) at x = a i.e., Remainder = f(a). |
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| Let p(x) be a polynomial divided by (x-a). |
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| Let q(x) be the quotient and R be the remainder. |
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| By division algorithm, |
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| Dividend = (Divisor x quotient) + Remainder |
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| p(x) = q(x) . (x-a) + R |
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| Substitute x = a, |
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| p(a) = q(a) (a-a) + R |
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| p(a) = R (a - a = 0, 0 - q (a) = 0) |
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| Hence Remainder = p(a). |
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| Example 1: |
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| Find the remainder when 2x3+4 is divided by |
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| a) x-1 |
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| b) x-2 |
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| d) x+1 |
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| Suggested answer: |
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| Let f(x) = 2x3 + 4 |
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| a) When f (x) is divided by x-1, |
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| R = f (1) |
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| = 2(1)3+ 4 |
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| = 6 |
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| b) When f (x) is divided by x-2, |
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| R = f (2) |
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| = 2(2)3+ 4 |
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| = 20 |
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| d) When f(x) is divided by x+1, i.e., x-(-1). |
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| R = f(-1) |
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| = 2(-1)3 + 4 |
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| = -2+4 = 2 |
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| Example 2: |
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| Factorise x2 - 7x + 10. |
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| Suggested answer: |
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| Let f(x) = x2 - 7x + 10 |
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| Then f (2) = (2)2 - 7(2) +10 [By trial and error method] |
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| To find the other factor of x2 - 7x + 10, divide x2 - 7x + 10 by (x-2) |
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| Verification: |
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| Example 3: |
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| Factorise x2 + 3x -4 using Remainder theorem. |
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| Suggested answer: |
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| Let f(x) = x2 + 3x - 4 |
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| Step 1: |
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| Find a value of the variable x for which the value of x2 + 3x - 4 |
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| is equal to 0. |
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| Let x =1, |
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| Then, f(1) = (1)2 + 3(1) - 4 = 0 |
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| Step 2: |
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| Divide x2 + 3x -4 by (x-1), |
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| Step 3: |
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| Example 4: |
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| Factorise 2x3 + x2 - 2x - 1. |
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| Suggested answer: |
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| Let f (x) = 2x3 + x2 - 2x - 1 |
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| = 0 |
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| Now, divide 2x3 + x2 - 2x - 1 by (x-1), |
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| 2x2 + 3x + 1 can be further factorised by splitting the middle term. |
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| The factors of 2x3 + x2 -2x -1 are (x -1) , (x+1) and (2x+1). |
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| Example 5: |
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| Factorise completely 2x3 - 7x2 - 3x + 18. |
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| Suggested answer: |
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| Let f (x) = 2x3 - 7x2 - 3x + 18 |
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| Let x = 1, |
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| This can be further factorised by splitting the middle term. |
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| Steps for factorisation using remainder theorem |
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 By trial and error method, find the factor of the constant for which the given expression becomes equal to zero. |
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 Divide the expression by the factor that is determined in step 1. |
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 Factorise the quotient. If the quotient is a trinomial, factorise it further. |
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 If the expression is a 4th degree expression, the first step will be to reduce this to a trinomial and then factorise this trinomial further. |
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