7.5 Roots and Zeros Objectives: The student will be able to…

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7.5 Roots and Zeros Objectives: The student will be able to… Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function.

Types of Roots A zero (aka root or solution) of a function f(x) is any value c such that f(c)=0. This means that (x-c) is a factor of f(x). the real zeros of the function are the x-intercepts of the graph.

Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. Imaginary numbers and real numbers both belong to the set of complex numbers. Solving polynomial equations may have one or more real roots, or no real roots (roots are imaginary numbers)

Example 1: Solve the equation. State the number and type of roots.

Example 2: Solve the equation. State the number and type of roots.

Example 3: Solve the equation. State the number and type of roots.

Example 4: Solve the equation. State the number and type of roots.

Corollary A polynomial equation of the form P(x) = 0 of degree n with complex coefficients has exactly n roots in the set of complex numbers.

Descartes’ Rule of Signs If P(x) is a polynomial with real coefficients whose terms are arranged in descending powers of the variable… -the number of positive real zeros of y=P(x) is the same as the number of changes in sign of the coefficients of the terms, or is less than this by an even number AND -the number of negative real zeros of y=P(x) is the same as the number of changes in sign of the coefficients of the terms of P(-x), or less than this by an even number.

Example 5: State the possible number of positive real zeros, negative real zeros, and imaginary zeros of

Example 6:

Complex Conjugates Theorem For any polynomial function, if an imaginary number is a zero of that function, its conjugate is also a zero.

Example 7: Write a polynomial function of least degree with integral coefficients that has -4, 1, and 5 as zeros.

Example 8: Write a polynomial function of least degree with integral coefficients whose zeros include 3 and 2-i