Polynomials and polynomial functions

1. Polynomials andpolynomial functions

2. Standard Form • Polynomial Function is classified by degree. • Its degree is the highest degree among its monomial term(s). • The degree determines the possible number of turning points in the graph and the end behavior of the graph

3. Linear Factors and Zeros • For any real number a and polynomial P(x), if x – a is a factor of P(x), then a is: • a zero of y = P(x) • a root ( or solution) of P(x) = 0 and • an x-intercepts of the graph of y = P(x) • If a is a multiplezero, its multiplicity is the same as the number of times x – a appears as a factor. • A turning point is a relative maximum or relative minimum of a polynomial function

4. Solving Polynomial Equations • Start by factoring • Use the Zero-Product Property to find solutions, or roots • The solutions may be real or imaginary. • Real solutions and approximations of irrational solutions can also be found by using a graphing calculator.

5. Dividing Polynomials • You can divide a polynomial by one of its factors to find another factor • Synthetic Division-dividing by a linear factor, you can simplify by writing only coefficients of each term.

6. Rational Root Theorem • A way to determine the possible roots of a polynomial equation P(x) = 0. • If the coefficients of P(x) are all integers, then every root of the equation can be written in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

7. Conjugate Root Theorem • If P(x) is a polynomial with rational coefficients, then irrational roots and imaginary roots of P(x) = 0 come in conjugate pairs. Therefore, if a + √b is an irrational root, where a and b are rational, then a - √b is also a root. Likewise, if a + bi is a root, where a and b are real and i is the imaginary unit, then a – bi is also a root.

8. Descartes’ Rule of Signs • A way to determine the possible number of positive and negative real roots by analyzing the signs of the coefficients. • the number of positive real roots is equal to the number of sign changes in consecutive coefficients of P(x), or is less than that by an even number. The number of negative real roots is equal to the number of sign changes in consecutive coefficients of P(-x), or is less than that by an even number

9. Fundamental Theorem of Algebra • If P(x) is a polynomial of degree n, where n ≥ 1, then P(x) = 0 has exactly n roots. This includes multiple and complex roots.