Remainder Theorem Let fx be an nth degree

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 • Reminder: f(k) is the remainder of f(x) divided by x – k.

Properties of Polynomials • An nth degree polynomial has n linear factors. Ex) f(x)= x 4 – 8 x³+ 14 x²+ 8 x -15 = ( x -1)(x+1)(x -3)(x-5) • An nth degree polynomial has n zeros. The zeros could be complex. Ex) f(x) = 2 x³ - 4 x² + 2 x 3 zeros 100 zeros Ex) f(x) = 3 x 100 + 2 x 85

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 - 4 x +5 If f(2 + i) = 0, then f(2 – i) = 0 • Ex) Let f(x) = x³ + 2 x² +x +2 f(i) = 0, f(-i) = 0, f( -2) = 0

Descartes Rule of Signs • Let f(x) be a polynomial of the form f(x) = anxn+an-1 xn-1+…. . a 1 x+a 0 1) 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. 2) 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.

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

Rational Zero Theorem • Let f(x) = anxn+an-1 xn-1+…. . a 1 x+a 0 • If f(x) has rational zeros, they will be of the form p/q, where • p is a factor of a 0 , and • q is a factor of an

Example • Find the list of all possible rational zeros for each function below. • A) f(x) = x³ + 3 x² - 8 x + 16 • B) f(x) = 3 x 4 + 14 x³ - 6 x² +x -12 • C) f(x) = 2 x³ - 3 x² + x – 6

Factoring for the finding Zeros of Polynomials • For 2 nd degree, we factored or used the quadratic formula. x² - 3 x – 10 = 0 , ( x – 5)(x + 2) = 0 so x = 5 or x = -2. For 3 rd degree, we factored. x³ - x² - 4 x + 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.

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!!