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Exploring Polynomial Equations Through the Fundamental Theorem of Algebra

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mpeltz
13 diapositives · 240 Views · Jan 04, 2020

Learn to use the Fundamental Theorem of Algebra to find polynomial equations with given roots, and understand how to identify all roots of a polynomial equation. Explore properties of real roots and write polynomial functions based on zeros.

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  1. 6-6 Fundamental Theorem of Algebra Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the roots of a polynomial equation. Holt Algebra 2

  2. You have learned several important properties about real roots of polynomial equations. You can use this information to write polynomial function when given in zeros.

  3. 2 3 8 11 3 3 P(x) = x3 – x2 – 2x + Example 1: Writing Polynomial Functions Write the simplest polynomial with roots –1, , and 4. If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial.

  4. Check It Out! Example 1a Write the simplest polynomial function with the given zeros. –2, 2, 4 If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial. P(x) = x3– 4x2– 4x + 16

  5. 11 2 3 3 P(x) = x3– x2+ 2x Check It Out! Example 1b Write the simplest polynomial function with the given zeros. 0, , 3 If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial.

  6. Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

  7. Example 2: Finding All Roots of a Polynomial Solve x4 – 3x3 + 5x2– 27x – 36 = 0 by finding all roots. The polynomial is of degree 4, so there are exactly four roots for the equation. p = –36, and q = 1. Graph y = x4 – 3x3 + 5x2– 27x – 36 to find the real roots. Find the real roots at or near –1 and 4.

  8. Example 2 Continued Test the possible real roots.

  9. Example 2 Continued The polynomial factors into (x + 1)(x– 4)(x2 + 9) = 0. The solutions are 4, –1, 3i,–3i.

  10. Check It Out! Example 2 Solve x4 + 4x3 – x2+16x – 20 = 0 by finding all roots. Find the real roots at or near –5 and 1.

  11. Check It Out! Example 2 Continued Step 2 Graph y = x4 + 4x3 – x2+ 16x – 20 to find the real roots. Find the real roots at or near –5 and 1.

  12. Check It Out! Example 2 Continued The polynomial factors into (x + 5)(x– 1)(x2 + 4) = 0. The solutions are –5, 1, –2i, +2i).

  13. HOMEWORK- Page 449 # 1-6, 11-19