Example of a polynomial function: f(x) = 4x^8 - 2x^4+ 7x^2
NOT examples of polynomial functions: f(x) = 3x^4.7 + 2x^2.1 f(x) = 6x^-3 + 2x^-2 + 4
Fundamental Theorem of Algebra -a degree n polynomial function will have n roots
To find the degree of a polynomial function you can usually count the changes of direction on it's graph.
This graph starts going in a negative direction, changes to positive, then goes negative, then ends going positive. Altogether that is four changes, so there are four root (where x=0) making it a fourth degree polynomial.
Not all graphs can be classified this way though. Some graphs have double, triple, quadruple, etc., roots. The graph below has a single, double, and triple root.
The first root on the left side is a single root because it only touches the x axis once. The second root is a double root because it "kisses" the x axis. The third one is a triple root because when examined closely, it touches the x axis three times.
A global maximum (shown above) is the highest point on the graph. A local maximum (shown above) is another high point on the graph, but it isn't as high as the global maximum. A global minimum is the lowest point on the graph. A local minimum is another low point on the graph, yet it isn't as low as the global minimum.
Some polynomial functions have imaginary roots. They always come in pairs of two. Imaginary roots can be found on a graph when there is a change of direction that doesn't result in the graph hitting the x axis. The graph below is an example of a third degree polynomial with two imaginary roots.
If a polynomial functions has a negative even leading coefficient, it will start in the third quadrant and end in the fourth quadrant. If a polynomial function has a negative odd leading coefficient, it will start in the second quadrant and end in the fourth quadrant. If a polynomial function has a positive even leading coefficient, it will start in the second quadrant and end in the first quadrant. If a polynomial function has a positive odd leading coefficient, it will start in the third quadrant and end in the first quadrant.
Polynomial Long Division In polynomial long division, you divide the divisor by the dividend. Then you make the leading term dividing the same as the leading term that is being divided by. You then continue to make the leading term dividing the same as the next number and so on until you have done them all and either have a remainder or non.
If you have a remainder, it is put over the divisor in the new equation, as shown above.
Synthetic Division Synthetic division can only be used when the divisor is a single root. Example: Synthetically divide 3x^3 + 8x^2 - 9x^2 + 2 by x - 1.
-First change the sign of the constant (-1 to 1). -Second place all the numbers into a line. Make sure there are all the powers in descending order. If the problem started with 3x^4 and didn't have anything to the power of three, then make sure to put a zero in the third power spot. -Then bring the three down and multiply it by 1. - Bring the product (3) to the next spot and add it to the 8 (11). -Multiply that by 1 and then add the product to -9. Repeat this until the end. The number in the last spot is the remainder. Final answer: (3x^3 + 8x^2 - 9x^2 + 2) / (x - 1) = 3x^2 + 11x + 2 + 4/(x-1).
Rational Root Theorem -Any rational roots from a polynomial are a quotient of factors from the constant and the leading coefficient Example: f(x) = 4x^3 - 4x^2 + 2 Factors of leading coefficient: 1, 2, 4 Factors of constant: 1, 2 Find the possible rational roots by putting the factors of the constant over the factors of the leading coefficient: +/- 1/1, +/- 2/1, +/- 1/2, +/- 2/2, +/- 1/4, +/- 2/4 Possible rational roots: +/- 1, +/- 2, +/- 1/2, +/- 1/4