1. IDENTIFY END BEHAVIOR OF A POLYNOMIAL FROM A GRAPH End behavior is the direction of the graph at the left and the right. There are four options for end behavior: up and up, down and down, down and up, and up and down. UP AND UP DOWN AND DOWN DOWN AND UP UP AND DOWN
2. IDENTIFY ZEROES OF A POLYNOMIAL FROM A GRAPH Zeroes of a polynomial are also known as its roots, solutions, or x-values of the x-intercepts. For instance, in the graph below, the polynomial would have zeroes of x = -3, -1, 1, and 3.
3. IDENTIFY TURNING POINTS OF A POLYNOMIAL FROM A GRAPH Turning points are any point on a graph where the polynomial changes direction, either from increasing to decreasing or decreasing to increasing. For instance, on the graph below, the polynomial would have turning points of (-1.5, -4), (0,0), and (1.5, -4).
4. IDENTIFY INTERVALS WHERE FUNCTION IS INCREASING AND DECREASING FROM A GRAPH If the graph has a negative slope, it is decreasing. On the other hand, if it has a positive slope, it is increasing. To write the intervals where a polynomial is increasing or decreasing, you will write the interval in the form (lowest x value of increasing/decreasing point, highest x value of increasing/decreasing point). Decreasing Intervals: (, 1. 5), (0, 1. 5) Increasing Intervals: ( 1. 5, 0), (1. 5, )
5. IDENTIFY LOCAL MAXIMUMS AND LOCAL MINIMUMS OF A POLYNOMIAL GRAPH Local maximums and local minimums will always be a turning point on a polynomial s graph. To determine if a turning point is a local max or local min, look at the points directly next to the turning points. If the turning point is higher than the points next to it, it is a local max. If it is lower than the points immediately next to it, it is a local min. Local Maximums: (0, 0) Local Minimums: ( 1. 5, 4), (1. 5, 4)
6. IDENTIFY END BEHAVIOR FROM A POLYNOMIAL EQUATION. The steps for identifying end behavior from an equation are shown below 1. Get the equation in standard form. Standard form means that terms are organized so that their exponents are arranged in descending order. An example is shown below. 4x 2 + 7x 5 2 + x 7x 5 + 4x 2 + x 2 2. Look at the leading term, which is the first term of the polynomial when written in standard form. For instance, in the equation above, the leading term is 7x 5 a. Leading term coefficient is positive and exponent is even: Up and Up b. Leading term coefficient is negative and exponent is even: Down and Down c. Leading term coefficient is positive and exponent is odd: Down and Up d. Leading term coefficient is negative and exponents is odd: Up and Down
7. IDENTIFY ROOTS/SOLUTIONS/ZEROES OF A POLYNOMIAL EQUATION. Roots, solutions, and zeroes are three different words that mean the same thing. When we find a root/zero of a polynomial, we are just finding the solution of the polynomial. There are several ways to find solutions to polynomials
8. Divide a polynomial by a factor using long division. Example: Divide (x 2 9x 10) by (x + 1) using long division.
9. DIVIDE A POLYNOMIAL USING SYNTHETIC DIVISION
10. CONVERT A POLYNOMIAL FROM FACTORED FORM TO STANDARD FORM. Example: Write (2x 7)(x + 1)(x 2 5) in standard form.
11. CLASSIFY A POLYNOMIAL BY DEGREE AND BY NUMBER OF TERMS The degree of a polynomial is equal to the highest exponent in the polynomial. For instance, the polynomial 4x 5 4x + 3 would have a degree of 5. The classifications by degree for polynomials are listed below. Degree of 0: Constant Degree of 1: Linear Degree of 2: Quadratic Degree of 3: Cubic Degree of 4: Quartic Degree of 5: Quantic We can also classify polynomials by the number of terms it has. Each term is separated by a plus or minus sign. For instance, the polynomial 5x 2 3x + 5 would have three terms. 1 Term: Monomial 2 Terms: Binomial 3 Terms: Trinomial 4 Terms: Polynomial of 4 Terms 5 Terms: Polynomial of 5 Terms
12. FIND THE MAXIMUM NUMBER OF TURNS IN THE GRAPH OF A POLYNOMIAL FUNCTION. We can determine the maximum number of turns of a graph from the degree of the polynomial. Simply subtract 1 from the degree of the polynomial and that is your maximum number of turns in the graph. Degree of Polynomial 1 = Max Turns in Graph Examples y = 3x + 5 has a degree of 1, and therefore has 0 max turns. y = 5x 5 + 3x 2 x + 99 has a degree of 5, and therefore has 4 max turns.
13. USE DESCARTES RULE OF SIGNS TO IDENTIFY POSSIBLE NUMBER OF POSITIVE AND NEGATIVE REAL ROOTS. The steps to use Descartes Rule of Signs are shown below. 1. Write the polynomial in standard form. 2. Identify the number of sign changes. Use the chart to identify the number of possible positive roots. 3. Plug in x for each x in the equation and simplify. Number of Sign Changes Possible Number of Roots 0 0 1 1 2 2 or 0 3 3 or 1 4 4 or 2 or 0 5 5 or 3 or 1 4. Count the number of sign changes again. Use the chart to state the number of possible negative roots. Example: What does Descartes Rule of Signs tell us about f(x) = 3x 2 4x + 2?
14. USE THE RATIONAL ROOT THEOREM TO IDENTIFY POSSIBLE RATIONAL ROOTS OF A POLYNOMIAL/SOLVE. 1. Write down all possible factors of the leading coefficient. Make sure to include positive and negative factors. 2. Write down all possible factors of the constant term. Make sure to include positive and negative factors. constant factors 3. Write down all possible combinations of leading coefficient factors 4. Test each possible root to see if it is a valid solution. Example: Solve x 3 27 = 0 by the Rational Root Theorem.
15. DESCRIBE TRANSFORMATIONS OF A POLYNOMIAL FROM ITS PARENT FUNCTION. A parent function is the simplest form of a polynomial. For instance, the parent cubic function is the equationy = x 3. Polynomial transformations are written in the form below Transformation Rules y = a(x + h) 3 + k 1. If a is negative, the parent function reflects across the x-axis. 2. If the absolute value of a is less than 1, the parent function is compressed by a factor of a. 3. If the absolute value of a is greater than 1, the parent function is stretched by a factor of a. 4. If h is added to x, the parent function is translated left by h units. 5. If h is subtracted from x, the parent function is translated right by h units. 6. If k is added to the end of the equation, the parent function is translated up by k units. 7. If k is subtracted at the end of the equation, the parent function is translated down by k units.
16. IDENTIFY DEGREE OF A POLYNOMIAL FROM A TABLE. Use the difference of differences rule to identify the degree of a polynomial from a table. You will subtract consecutive y values until and write the results until you have reached set of values that are all equal to each other. An example is shown below. x y -2 15-1 0 0-1 1 0 2 15
17. IDENTIFY FACTORS FROM ZEROES/ROOTS. Factors can easily be written from roots/solutions. All you do is change the sign of the root and add it to x. For instance, a root of -4 would give a factor of (x+4). A root of 2 + i would result in a factor of (x-(2+i)), which can also be written as (x-2-i). The Complex Conjugate Root Theorem This theorem tells us that if a complex number is a root of a polynomial, its conjugate is also a root. Examples are listed below. A polynomial with a root of 3+i would also have a root of 3-i A polynomial with a root of -4i would also have a root of 4i A polynomial with a root of 2-6i would also have a root of 2+6i