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7.5 and 7.6 Roots and Zeros of a Function

7.5 and 7.6 Roots and Zeros of a Function. Zeros, Factors & Roots Summary:. c is a zero of f(x) x - c is a factor of f(x) c is a root/solution of f(x) = 0 If c is real, (c,0) is an x- intercept. Fundamental Theorem of Algebra.

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7.5 and 7.6 Roots and Zeros of a Function

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  1. 7.5 and 7.6 Roots and Zeros of a Function

  2. Zeros, Factors & RootsSummary: • c is a zero of f(x) • x - c is a factor of f(x) • c is a root/solution of f(x) = 0 • If c is real, (c,0) is an x-intercept.

  3. Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex #’s. An nth degree polynomial equation of the form P(x) = 0 • has exactly n roots in the set of complex #’s. • has exactly n zeros.

  4. Rational Zero Theorem • can help us limit the # of possible zeros to test • p is a factor of the constant • q is a factor of the leading coefficient

  5. Ex 1List all the possible rational roots. • f(x) = x4 + 7x3 – 15 • g(x) = 6x3 + 6x2 – 15x - 2

  6. EX 2Find all the zeros. • f(x) = x4 + x3 - 19x2 + 11x + 30

  7. Complex Conjugates Theorem if a + bi is a zero, then a - bi is a zero Irrational Root Theorem

  8. EX 3Find all the zeros. • f(x) = x3– 4x2 + 6x – 4

  9. EX 4Find all the zeros. • f(x) = 3x3 - 5x2 - 14x - 4

  10. EX 5Find all the zeros. • f(x) = 6x4– 5x3 - 38x2 - 5x + 6

  11. EX 6Find all the zeros. • f(x) = 6x4– 17x3 + 8x2 + 8x - 3

  12. Ex 7Write the polynomial function of least degree w/integer coefficients whose zeros include the following: Example 5-1c

  13. Ex 8Write the polynomial function of least degree w/integer coefficients whose zeros include the following: Example 5-1c

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