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Exploring Real Zeros in Algebra II: Fundamental Theorem Applications

Dive into the concept of real zeros and the Fundamental Theorem of Algebra in this comprehensive guide. Learn how to apply Descartes' Rule of Signs and find all real zeros of polynomial functions. Practice solving equations and discovering the roots of various polynomial functions. Get ready for the test by mastering key theorems and techniques.

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Exploring Real Zeros in Algebra II: Fundamental Theorem Applications

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  1. Do Now: Find all real zeros of the function.

  2. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9: Tuesday, 12/2

  3. Descartes’ Rule of Signs • The number of positive real zeros of f is equal to the number of sign changes in the sign of the coefficients of f(x) or is less than this by an even number. • The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.

  4. Fundamental Theorem of Algebra • If f(x) is a polynomial with degree of n (where n>0), then the equation f(x) = 0 has at least one solution. • Corollary: The equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

  5. Fundamental Theorem of Algebra (x+3)(x – 4)2= 0 • Example: x3 – 5x2 – 8x + 48 = 0 x = -3 and x = 4 This equation only has two distinct solutions: -3 and 4. Because the factor (x – 4) appears twice, you can count the solution 4 twice. With the repeated solution the third-degree equation has three solutions.

  6. Solve the equation. x3 + 5x2 + 4x + 20 = 0

  7. Find the zeros of the polynomial function. f(x) = x4 – 8x3 + 18x2 – 27

  8. Find all zeros of the polynomial function. f(x) = x5 – 2x4 + 8x2 – 13x + 6

  9. Conjugates Theorem • If is an imaginary zero of function f, then is also a zero. • If is a zero of function f where is irrational, then is also a zero.

  10. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, & the given zeros. 1.) -1, 2, 4 2.) 4,

  11. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, & the given zeros. 3.) 2, 2i,

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