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Math Analysis

Math Analysis. 2.5 The Fundamental Theorem of Algebra. Objectives. 1) Understand and use the FToA 2) Find all the zeros of a polynomial function. 3) Write a polynomial with real coefficients given its zeros. Fundamental Theorem of Algebra.

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Math Analysis

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  1. Math Analysis 2.5 The Fundamental Theorem of Algebra

  2. Objectives • 1) Understand and use the FToA • 2) Find all the zeros of a polynomial function. • 3) Write a polynomial with real coefficients given its zeros.

  3. Fundamental Theorem of Algebra • If f(x) is a polynomial of nth degree, then it has n solutions. • These solutions can be complex or real. • Solutions can be repeated.

  4. Rational Zero Test • If a polynomial has integer coefficients, every zero has the form p/q where p is a factor of the constant and q is a factor of the leading coefficient. • Example 1: Find the rational zeros of • f(x)=x3-4x2-4x+16

  5. Conjugate Pairs • If a+bi where b≠0 is a solution, then a-bi is also a solution. • Example 2: Find a polynomial with real coefficients of 4th degree with the given solutions: 4, -2, 3i

  6. Example 3: Finding Zeros • Use the given zero to find all the zeros of the function. • f(x)= x3-7x2-x+87, 5+2i

  7. Factoring a Polynomial • A quadratic factor with no real zeros is said to be prime or irreducible over the reals.

  8. Example 4 • Find all the zeros and write the polynomial as a product of linear factors. • x4 +6x3+10x2+6x+9

  9. assignment • Page 181: 1-89 odds, 93, 100

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