A3 3.4 Zeros of Polynomial Functions Homework: p eoo, odd

Rational Zeros Theorem Real zeros of polynomial functions are either rational zeros or irrational zeros. Examples: The function has rational zeros –3/2 and 3/2 The function has irrational zeros – 2 and 2

Rational Zeros Theorem Suppose f is a polynomial function of degree n > 1 of the form with every coefficient an integer and. If x = p /q is a rational zero of f, where p and q have no common integer factors other than 1, then p is an integer factor of the constant coefficient, and q is an integer factor of the leading coefficient.

RZT – Examples: Find the rational zeros of The leading and constant coefficients are both 1!!!  The only possible rational zeros are 1 and –1…check them out: So f has no rational zeros!!! (verify graphically?)

RZT – Examples: Find the rational zeros of Potential Rational Zeros: Factors of –2 Factors of 3 Graph the function to narrow the search… Good candidates: 1, – 2, possibly –1/3 or –2/3 Begin checking these zeros, using synthetic division…

RZT – Examples: Find the rational zeros of 134–5– Because the remainder is zero, x – 1 is a factor of f(x)!!! Now, factor the remaining quadratic… The rational zeros are 1, –1/3, and –2

RZT – Examples: Find the polynomial function with leading coefficient 2 that has degree 3, with –1, 3, and –5 as zeros. First, write the polynomial in factored form: Then expand into standard form:

RZT – Examples: Using only algebraic methods, find the cubic function with the given table of values. Check with a calculator graph. x–2–115 f(x) (x + 2), (x – 1), and (x – 5) must be factors… But we also have :

Properties of Roots of Polynomial Equations 1.A polynomial equation of degree n has n roots, counting repeated roots separately 2.If a+bi is a root to the polynomial equation with real coefficients ( ), then the imaginary number a-bi is also a root. Imaginary roots, if the occur, always occur in conjugate pairs

Theorem: Linear Factorization Theorem If f (x) is a polynomial function of degree n > 0, then f (x) has precisely n linear factors and where a is the leading coefficient of f (x), and are the complex zeros of f (x). the are not necessarily distinct numbers; some may be repeated.

Fundamental Polynomial Connections in the Complex Case The following statements about a polynomial function f are equivalent even if k is a nonreal complex number: 1. x = k is a solution (or root) of the equation f (x) = k is a zero of the function f. 3. x – k is a factor of f (x). One “connection” is lost if k is a complex number… k is not an x-intercept of the graph of f !!!

and now for some cool little theorems… Fundamental Theorem of Algebra: if f(x) is a polynomial of degree n, then the equation f(x)=0 has at least one complex root. Linear Factorization Theorem: given then the linear factorization is: Notice: the a’s are the same, and the linear factors are the zeros…

some examples… 1. Find a 4 th degree polynomial function with real coefficients that has zeros of -2, 2, and i such that f(3)=-150. Write the equation in factored form, and in general form. 2. Find an n-th degree polynomial function with real coefficients. Write the complete linear factorization and the polynomial in general form. n=3 (degree), x=6 and – 5+2i, f(2)=- 636