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Section 5.5– Zeros of Polynomial Functions

Section 5.5– Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. Every polynomial function of degree n, with , has at least one zero in the system of complex numbers. Result of the Fundamental Theorem.

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Section 5.5– Zeros of Polynomial Functions

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  1. Section 5.5– Zeros of Polynomial Functions

  2. The Fundamental Theorem of Algebra Every polynomial function of degree n, with , has at least one zero in the system of complex numbers. Result of the Fundamental Theorem Every polynomial function f of degree n, with , can be factored into n linear factors (not necessarily unique).

  3. Find a Polynomial Function Find a polynomial function of degree 3 with the given numbers as zeros. Find the coefficient using the point .

  4. Occurrence of Zeros For Polynomial Functions with Real Coefficients: • Imaginary zeros occur in conjugate pairs. For Polynomial Functions with Rational Coefficients: • Zeros containing square roots occur in pairs. If you know that the polynomial has a zero of , then there MUST be another zero: If you know that the polynomial has a zero of , then there MUST be another zero:

  5. Occurrence of Zeros Suppose that a polynomial function of degree 4 with rational coefficients has the given numbers as zeros. Find the other zeros(s).

  6. Zeros of Polynomials Find the zeros. Try factoring!

  7. Zeros of Polynomials

  8. Rational zeros theorem Let , where all the coefficients are integers. Consider a rational number denoted by , were p and q are relatively prime (having no common factors besides -1 and 1). If is a zero of , then p is a factor of and q is a factor of . List all possible rational zeros of the function.

  9. Find zeros • Solve • Factor into linear factors Which one should we start with? Look at the graph! It appears that and are zeros. Let’s see if and are factors.

  10. Find zeros Use synthetic division to see if is a zero and is a factor. Use synthetic division to see if is a zero and is a factor.

  11. Find zeros Use the quadratic formula to find the last two zeros.

  12. Descartes’ Rule of Signs • Let be a polynomial function with real coefficients • The number of positive real zeros is equal to the number of sign changes of or less than that by an even number. • 7 sign changes or positive real root(s) • 12 sign changes or positve real root(s) • The number of negative real zeros is equal to the number of sign changes of or less than that by an even number. • 3 sign changes or negative real root(s) • 8sign changes or negative real root(s)

  13. Descartes’ Rule of Signs 7. Use Descartes’ Rule of Signs to determine the possible number of positive and negative number of solutions. 2 or 0 positive real roots 1 negative real root

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