Polynomials and Polynomial Functions

1. Polynomials and Polynomial Functions Chapter 6

2. Things we’ll study … • What’s the chapter about • Polynomials • Polynomial Functions – how to evaluate, graph and find the zeros • Polynomial Equations – How to perform operations on and solve • Properties of exponents • Scientific Notation • Synthetic substitution • Graph and investigate end-behavior of polynomial functions • Add, subtract, multiply and divide polynomials • Use factoring and synthetic division to find the rational zeros of a polynomial function • The Fundamental Theorem of Algebra • Rational Zero Theorem

3. Sec 6.1 Using Properties of Exponents • Definition of a power • The 7 properties of exponents • What they are • Evaluating numerical expressions • Simplifying algebraic expressions • Scientific Notation

4. Sec 6.2 Evaluating and Graphing Polynomial Functions • Polynomial functions • Standard form • Leading coefficient • Degree, name • Evaluating polynomial functions • Direct substitution • Synthetic substitution • Investigating end behavior of a polynomial function • How does the function behave as x approaches infinity/negative infinity

5. End Behavior • For

6. Sec 6.3 Adding, Subtracting and Multiplying Polynomials • Adding/Subtracting Polynomials • Horizontally or Vertically • Vertically – line up common terms • Horizontally – collect common terms • Multiplying Polynomials • Horizontally or Vertically • Cube of a Binomial • Special Pattern

7. Sec 6.4 Factoring and Solving Polynomial Equations • Factoring Polynomials • Special factoring patterns • Factor by grouping • Quadratic form • Solving Polynomial Equations • Zero Product Property

8. Sec 6.5: The Remainder and Factor Theorems • Dividing polynomials using long division • Using synthetic division when the divisor is ‘x-k’ • Remainder Theorem • If a polynomial is divided by x-k then the remainder r = f(k) • A closer look at synthetic division – the quotient • Factor Theorem • A polynomial f(x) has a factor x-k if and only if f(k) = 0.

9. Let’s Summarize … • We can evaluate a polynomial using synthetic substitution (6.2) • When we solve polynomial equations, we factor and then use the zero product property to find the zeros (6.4) • We can do polynomial long division (6.5) • We know if an expression goes into another expression evenly ie) no remainder, then it is a factor of that expression • If a divisor of a polynomial is of the form ‘x-k’, then we can use synthetic division to find the quotient and remainder. Synthetic division looks just like synthetic substitution (6.6) That led us to the Remainder Theorem • The Factor Theorem tells us if f(k) = 0, then x-k is a factor (6.6) • So if we are given a polynomial and one of it’s zeros, or • Know a value of x for which the polynomial equals zero, then • We can factor the polynomial

10. Sec 6.6: Finding Rational Zeros • Rational Zero Theorem • If f(x) is a polynomial with integer coefficients, then every rational zero of f has the form p is a factor of the constant term q is a factor of the leading coefficient

11. Sec 6.6How will we use this theorem? • So given just the polynomial equation, and • No other information • We can • Find all the rational zeros • Find all the real zeros • Now you can see it’s important to distinguish between real numbers and rational numbers

12. Sec 6.7 Using the Fundamental Theorem of Algebra • In general, the Fundamental Theorem of Algebra says when all real and imaginary solutions are counted, an nth degree polynomial has exactly n solutions/roots • Review: Real, Complex and Imaginary numbers • Some examples • Interesting things • Real zeros intersect the x-axis – imaginary zeros don’t • When a real factor is raised to an odd power, it crosses the x-axis • When a real factor is raised to an even power, the graph is tangent to the x-axis • Complex zeros always occur in pairs • What do you do when the coefficients of the polynomial are not integers ? Use your graphing calculator.