1. What is a Polynomial Function?
1.1. A polynomial function is any function that contains a polynomial expression in one variable.
1.1.1. Example
2. Dividing Polynomial Functions
2.1. Long Division
2.1.1. With long division, there is no restriction with dividing polynomials unlike Synthetic Division. For example you can divide "4x^3-7x^2-11x+5" and "x^4+6x^2+2" no problem but using long division for any polynomial that has a coefficient greater than 1 is a waste of time, that's why we would use synthetic for those type of polynomials.
2.1.1.1. Example
2.2. Synthetic Division
2.2.1. In order to divide polynomials using synthetic division, you must be dividing by a linear expression and the leading coefficient must be a 1. For example, you can use synthetic division to divide "4x^3-7x^2-11x+5" but you cannot use it to divide "x^4+6x^2+2", we would use long division.
2.2.1.1. example
3. Remainder Theorem
3.1. When a polynomial, f(x) is divided by x - a, the remainder is equal to f(a). If the remainder is zero, then x - a is a factor of the polynomial. This can be used to help factor polynomials.
3.1.1. Example
4. Factor Theorem
4.1. The factor theorem is a special case of the remainder theorem which states that if f(a) = 0, then x - a is a factor of the polynomial f(x). Factor Theorem should be used for factoring polynomials with a higher degree than 2
4.1.1. Example
5. Vocalbulary
5.1. Monomial - a number/product of numbers and variables with whole number exponents.
5.1.1. Polynomial - a monomial or a sum or difference of monomials.
5.1.1.1. Degree of a polynomial - the degree of the first term indicates it.
5.2. Leading coefficient - the coefficient of the first term.
5.2.1. Binomial - polynomial with two terms.
5.2.1.1. Trinomial - has three terms.
6. Multiplying Polynomials
6.1. To multiply a polynomial by a monomial use the Distributive Property and the Properties of Exponents
6.1.1. Example
6.2. To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first.
6.2.1. Example
7. End Behaviours
7.1. The end behaviour of a function f describes the behaviour of the graph of the functions "ends" of the x-axis.
7.1.1. The degree and the leading coefficient in the equation of a polynomial function indicate the end behaviours of the graph.
7.1.1.1. For Example
7.1.1.2. Understanding the chart above can help indicate the end behaviours of the graph. For example. "2x^5-4x^3+10x^2-13x+8" We see that the leading coefficient is positive but our degree is odd. Now we can determine our end behaviours "x -> -∞, g(x) -> -∞ and x -> ∞, g(x) -> ∞".
7.2. End Behaviours are also seen in Calculus, called "Limit Notation" or "Lim f(x)". Limit Notation is used to describe the end behaviours.
8. Turning Points
8.1. A turning point of a function is a point where the graph changes from increasing to decreasing or the other way around. Curves of the highest degree of "n" can have up to (n - 1) turning points.
8.1.1. Example
8.1.2. We can determine the amount of turning points in our function. Using (n-1) we can find the turning points. "f(x)= x^4-6x^3+3x^2+10x-3" Using (n-1), we can get the turning points. ("4"-1) = 3. We can determine that the function will have 3 turning points.
8.2. Turning points are seen in Calculus called stationary points. Stationary points are found by using a method called "Differentiation"
9. Roots/Zeros
9.1. Possible Root
9.1.1. Root or Zeros of a polynomial function are values of x that cause the polynomial to =0. Which is a X-intercept. Odd degree functions must have at least 1 root. Even degree function may have 0 roots. Degree of N may have up to N distinct roots.
9.1.1.1. example
9.1.1.1.1. We can determine how many roots a polynomial function may have. By looking at the highest degree of the polynomial. Ex. "2x^3-4x^2+3x+2" The highest degree is 3, therefor the number of possible roots is 3.
9.2. Specific Root
9.2.1. To find specific root, you can use synthetic division to calculate your specific root or you can factor.
9.2.1.1. example of using synthetic
9.2.1.2. example of factoring
10. Even/Odd degree
10.1. We write a polynomial so that terms go from highest to lowest order. The degree of the polynomial is the highest order. So F(x) = 5x^2-2x+4 has a degree of 2 and g(x) = -75x^5-3x^4+4x has a degree of 5. If a polynomial has a odd degree, then it's called "odd degree polynomial", if it's even then it's called "even degree polynomial"
10.1.1. example