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Remainder Theorem

Remainder Theorem . Let f(x) be an nth degree polynomial. If f(x) is divided by x – k, then the remainder is equal to f(k). We can find f(k) using Synthetic Division. Factor Theorem . If f(k)= 0 , then x – k is a factor of f(x). If x – k is a factor of f(x), then f(k) = 0

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Remainder Theorem

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  1. Remainder Theorem • Let f(x) be an nth degree polynomial. • If f(x) is divided by x – k, then the remainder is equal to f(k). • We can find f(k) using Synthetic Division.

  2. Factor Theorem • If f(k)= 0, then x – k is a factor of f(x). • If x – k is a factor of f(x), then f(k) = 0 • Reminder: f(k) is the remainder of f(x) divided by x – k.

  3. Properties of Polynomials • An nth degree polynomial has n linear factors. Ex) f(x)= x4 – 8x³+ 14x²+ 8x -15 = ( x -1)(x+1)(x -3)(x-5) • An nth degree polynomial has n zeros. The zeros could be complex. Ex) f(x) = 2x³ - 4x² + 2x 3 zeros Ex) f(x) = 3x100 + 2x85 100 zeros

  4. Conjugate Pairs Theorem • Let f(x) be an nth degree polynomial with real coefficients. • If a+bi is a zero of f(x), then the conjugate a – bi must also be a zero of f(x). • Ex) Let f(x) = x² - 4x +5 If f(2 + i) = 0, then f(2 – i) = 0 • Ex) Let f(x) = x³ + 2x² +x +2 f(i) = 0, f(-i) = 0, f( -2) = 0

  5. Descartes Rule of Signs • Let f(x) be a polynomial of the form f(x) = anxn+an-1xn-1+…..a1x+a0 • The number of positive real zeros of f(x) is equal to the number of sign changes of f(x) or is less than that number by an even integer. • The number of negative real zeros of f(x) is equal to the number of sign changes in f(-x) or is less than that number by an even integer.

  6. Example • Find all possible positive, negative real and nonreal zeros of f(x) = 4x4- 3x³ +5x² + x – 5

  7. Rational Zero Theorem • Let f(x) = anxn+an-1xn-1+…..a1x+a0 • If f(x) has rational zeros, they will be of the form p/q, where • p is a factor of a0 , and • q is a factor of an

  8. Example • Find the list of all possible rational zeros for each function below. • A) f(x) = x³ + 3x² - 8x + 16 • B) f(x) = 3x4 + 14x³ - 6x² +x -12 • C) f(x) = 2x³ - 3x² + x – 6

  9. Factoring for the finding Zeros of Polynomials • For 2nd degree, we factored or used the quadratic formula. x² - 3x – 10 = 0 , ( x – 5)(x + 2) = 0 so x = 5 or x = -2. For 3rd degree, we factored. x³ - x² - 4x + 4 = 0 , x²(x -1) -4(x – 1) =0 ( x – 1)(x² - 4) = 0 , (x – 1)(x - 2)(x + 2) =0 x = 1, x = 2, x = -2 But, Factoring by traditional means doesn’t always work for all polynomials.

  10. Strategy for Finding all the zeros of a Polynomial • Step 1: Use Descartes Rule of Signs • Step 2: Use Rational Zeros Theorem to get list of possible rational zeros. • Step 3: From the list above, test which ones make f(x) = 0. • Do this using SYNTHETIC DIVISION!!!! • Do not plug in the values into f(x)!!! • We want to factor f(x) until we get a quadratic function. Check Mate!!

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