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Mastering the Remainder Theorem in Polynomial Equations

Learn how to find factors, roots, and remainders in polynomials using the Remainder Theorem. Understand f(x) notation and factorizing techniques. Practice exercises provided.

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Mastering the Remainder Theorem in Polynomial Equations

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  1. Lesson 4-Remainder Theorem Objectives : - The remainder and Factor theorems - It’s used to help factorise Polynomials - It’s used to help find roots of equation - It’s used to help find remainder without actually dividing ML4 MH

  2. f(x) Notation • f(x) stands for “factor of x” • f(x) notation is often used instead of “y” • So we can say f(0) = 03-7(0)2+5(0)-6 = -6 f(1) = 13-7(1)2+5(1)-6 = -7 f(2) = 23-7(2)2+5(2)-6 = -16 f(-1)= (-1)3-7(-1)2+5(-1)-6 = -19 ML4 MH

  3. Consider this division x-2 is a factor of x3-3x2-10x+24 Calculate f(2) f(2)=0 • __x2-x-12____ • (x-2) | x3-3x2-10x+24 • x3-2x2 • -x2-10x • -x2+ 2x • - 12x+24 • - 12x+24 • 0 If f(a)=0, then (x-a) is a factor of the Polynomial so if f(a) = 0 the remainder is zero ML4 MH

  4. Consider this division Calculate f(-2) What do you notice f(-2)= 2(-2)3+3(-2)2-(-2)+1 f(-2)=-1 If a polynomial f(x) is divided by (x – a), the remainder is the constant f(a), and f(x) = q(x) ∙ (x – a) + f(a) where q(x) is a polynomial with degree one less than the degree of f(x). Why is this ?? ML4 MH

  5. This is because A division is essentially : When x=a i.e. f(a) if f(a)=0 -> factor (x-a) otherwise f(a)=R -> (x-a) gives R Divisor d(x) is of the form (x-a) Do exercise 3c page 94 ML4 MH

  6. Summary • If a polynomial f(x) is divided by (x – a), • the remainder is the constant f(a), and • f(x) = q(x) ∙ (x – a) + f(a) • where q(x) is a polynomial with degree • one less than the degree of f(x). ML4 MH

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