15-2 Graphs and Zeros of Polynomial Functions
Alright....APs are over so now it's time to buckle down and focus on Chapter 15!
This section shows how to recognize the degree of a polynomial function and how to find the zeros.
Stuff you should know:
Zero of a Function: A zero of a function f is an x-value c for which f(c)=0.
Example: f(x)=x^2 -6x+5 = (x-5)(x-1) *by factoring*
So...the zeros equal 5 and 1.
The degree of a one-variable polynomial is the same as the greatest exponent of the variable.
Synthetic substitution is a quick way to evaluate a polynomial function. (It's easier than long division of polynomials!!)
Synthetic substitution can be used to find out if the zeros are accurate.
~Put the zero that we want to try into a little box.
~Write the coefficients of the equation next to the box.
~Write a line under the coefficients, leaving enough room for another number.
~Drop the first coefficient below the line.
~Multiply the number in the box with the number you just put below the line.
~Put the product below the second coefficient.
~Add the coefficient and the product and put the sum below the line.
~Multiply the sum with the number in the box and repeat the steps.
If the last number behind the line is a zero, then the number in the box is a zero of the function.
The remaining numbers behind the line are the coefficients of the other factor of the equationThings to know for Synthetic Substitution:
*If the last number under the line is not 0, then the x-value is not a zero for the function. The number that you get in the last spot is called the remainder.
*The remaining numbers below the line are the coefficients for the other factor of the function. The factor is always one degree less than the degree of the function you started with. So, if you start with a cubic function, the factor you get with synthetic division is quadratic.
*The Remainder Theorem: If p(x) is a polynomial, then p(c) equals the remainder when p(x) is divided by the quantity (x-c).
*The Factor Theorem: (x-c) is a factor of polynomial p(x) if and only if p(x)=0.
The Fundamental Theorem of Algebra and Its Corollaries: A polynomial has at least one zero in teh set of complex numbers.
Corallary: An nth-degree polynomial function has exactly n zeros in the set of complex numbers, counting multiple zeros.
Corollary: If a polynomial only has only real coefficients, then any nonreal complex zeros appear in conjugate pairs.Translation: You always have the same number of zeros as the degree of the polynomial (if you count both complex and real numbers). If you have an imaginary zero, it will have a pair.
A cubic function will always have three zeros, but two of them could be complex.
Sums and Products of Zeros:
Sums of the Products of the Zeros of a cubic function:
If p(x)=ax^3+bx^2+cx+d has zeros z1, z2, and z3, then--
Sum of the zeros: z1+z2+z3=-b/a
Sum of the pariwise products of the zeros: z1(z2)+z1(z3)+z2(z3)=c/a
Product of the zeros: z1*z2*z3=-d/a
The sum of the zeros usually fills the B spot of a cubic function, the sum of the pairwise usually fills the C spot, and the product usually fills the D spot of a cubic function.
The values of the coefficients for the cubic functions are opposite for the sum and product of the zeros.
Example: 1)Find the sum of the zeros, the sum of the pairwise products, and the products of all three zeros if the z1=.6, z2=2 and z3=4.
2)Find the particular equation of the cubic function with values you find. 3)Prove the sum, sum of pairwise products, and products are correct by using the sum and products property of the zeros of a cubic function.
Sum of Pairwise: (.6)(2)+(.6)(4)+(2)(4)=11.6=58/5
2) Because you know what places the values take in the cubic function, you can figure out that the particular equation of one of the possible functions is y=x^3-(33/5)x^2+(58/5)x-(24/5).
Since the coefficients of the values are all over 5, we can change the coefficients to be integers.
So: f(x)=5(x^3)- 33(x^2)+58x-24
3) To prove the values we got in number 1 are correct, we can use the properties of this section:
Sum: (-b/a)=(33/5) ~it checks out
Sum of Pairwise: (c/a)=(58/5) ~it checks out
Product: (-d/a)=(24/5) ~it checks out
(thank Leo for this part)
~Low point = critical point = extreme point = vertex
~Number of extreme points = the power – 1
Example: Cubic functions have 2 extreme points
~Odd powers go in two different directions
~In even number powers, the -/+ on the first term determines the up and the down
~At the Point of inflection, the concavity changes.
Definition of concavity:
Concave down = tangent lines are above
Concave up = tangent lines are below
That's pretty much it!
For extra help: http://www.purplemath.com/modules/synthdiv.htm
**Gina....you're up next!
Everyone should be excited for two really cool movies that are coming out in the next two weeks:
Shrek the Third:And:
Pirates of the Caribbean: At World's End