Polynomial Function
A function that can be written as f(x) = a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x+a_{0}, where coefficients are real numbers.
A function made of terms with powers of x.
Ex: f(x) = 2x^3 - 3x + 1
Degree of a Polynomial
The highest power of x in a polynomial; tells the number of roots possible.
The biggest exponent shows how many roots there could be.
Ex: 4x^3 - 3x^2 + 7 -> degree 5
Root (Zero)
A value of x that makes f(x) = 0.
A number that makes the function equal zero.
Ex: f(2) = 0 -> x = 2
Real Root
Root found on the real number line; appears where the graph crosses the x-axis.
Where the graph touches or crosses the x-axis.
Ex: f(x) = (x-2)(x^2+1) -> x = 2
Imaginary Root
Root that involves i=\sqrt{-1}; always comes in pairs.
Root that doesn’t show up on the graph - use i.
Ex: x = (+-)i for x^2 + 1 = 0
Complex Conjugate Pair
If a + bi is a root, a - bi is also a root (for real coefficents).
Imaginary roots always come in pairs.
Ex: 2 +3i & 2 - 3i
Rational Root Theorem
Lists possible rational roots as p/q: p = factors of constant, q = factors of leading coefficent.
Rule for finding fraction or whole number roots.
Ex: 2x^3 - 3x^2 + 2x - 1 -> (+-)1, (+-)1/2
Integral Coefficents
Coefficents that are integers (no fractions or decimals).
Whole number coefficents.
Ex: f(x) = x^3 - 5x^2 + 8x - 4
Synthetic Division
Shortcut for dividing by a linear factor x - c; tests possible roots.
Quick way to check if a number iis a root.
Ex: Divide f(x) by x-2
Factor Theorem
x - c is a factor if f(c) = 0
If plugging in gives zero,it divides evenly.
Ex: f(3) = 0 -> (x - 3) is a factor
Fundamental Theorem of Algebra
Every polynomial of degree n has exactly n roots (real or imaginary).
Numbers of roots = the degree of the polynomial.
Ex: Cubic -> 3 total roots
Multaplicity of a Root
The number of times a root appears. Odd = crosses x-axis, Even = touches only.
How many times a root repeats.
Ex: f(x) = (x - 2)^2(x + 1)
Zero Root
A root at x = 0; polynomial includes and x factor.
When zero makes the function equal zero.
Ex: f(x) = x(x - 2)(x+3)
Possible Root Combinations
Lists all possible counts of positive, negative, zero, and imaginary roots.
All possible ways the roots can appear.
Ex: Degree 4 -> 4, 2, 0 positive; 0, 2, 4 negitave
Graphical Analysis of Roots
Using a graph to find x-intercepts (real roots).
Look at the graph to real roots.
Ex: Graph crosses x-axis 2 times
Irrational Root
Real root that can’t be written as a fraction intergers.
A decimal that never ends or repeats.
x=\sqrt{2}
Analytical Analysis of Roots
Using alebraic methods (factoring, formulas, division) to find roots.
Solving with equations instead of guessing.
Ex: Use quadratic formula or synthetic division
