Chapter 11: Polynomials

Chapter 11 Preveiw: aach11preview.pdf
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Wiki Summary Assignments

11-1: Max S. and Kaitlynn R.
11-2: Nate F. and Caiti T.
11-3: Justin F. and Ricky S.
11-4: Austin R. and Shannon S.
11-5: Elissa M. and Shannon M.
11-6: Nate F. and Ricky S.
11-7: Thomas S. and Caiti T.
11-8: Brittany K. and Max. S.
11-9: Marisa M. and Shannon S.
11-10: Sarah K. and Austin R.

Section 11-1: Introduction to Polynomials

Notes: aanotes11-1.pdf

To determine a degree of a polynomial look at the equation and find the largest exponent. The terms of the polynomial are a collection of variable and coefficients in the polynomial: seperated by + or -. To write a polynomial in standard form all you have to do is write it in descending order of it's coefficients. The leading coefficient is the number that in written with the variable of the highest degree. For example if we have the equation 35x³ + 25x² + 100, the degree would be 3, and the leading coefficient would be 35.
Katie R.

Max J. S.
In this lesson we learned about polynomials. We learned the the degree of a polynomial is the largest exponent of the variable. We also learned that each collection of a variable and coefficient in the polynomial aProxy-Connection: keep-alive Cache-Control: max-age=0 the terms of the polynomial. The standard form of a polynomial is each term in descending order of degree. The leading coefficient is the number with the variable of highest degree. We learned about some special types of polynomials such as, linear, quadratic, cubic, and quartic. A polynomial function is a function where P(x) where P(x) is a polynomial.

Section 11-2: Polynomials and Geometry

Notes: aanotes11-2.pdf

Monomial- Polynomial with one term.
Binomial- Polynomial with two terms.
Trinomial- Polynomial with three terms.
When a polynomial of one variable is added or multiplied by another polynomial, there must be the degree of a polynomial in several variables.
The extended distributive property- To multiply two polynomials, multiply each term in the first polynomial by each term in the second.
Nate F.

The vocab in this section was fairly simple. We learned that a monomial is a polynomial with one term, a binomial is a polynomial with two terms, a trinomial is a polynomial with three terms and a monomial is a polynomial a higher amount of terms than three. The extended distributive prop is multiplying two polynomials by the first and second terms. There is a must be the degree of a polynomial in several variables when adding or multiplying a polynomial of one variable by another polynomial.
Caiti T.

Section 11-3: Factoring Special Cases

Day 1

Notes: aanotes11-3(1).pdf

Justin F.- Factoring is rewriting a polynomial as a product of factors. 1. Greatest common factor 2. Binomial square factoring 3. Difference of squares factoring 4. Other methods of factoring
Binomial Square Factoring- For all a and b: a^2+2ab+b^2=(a+b)^2 and a^2-2ab+b^2=(a-b)^2
Difference of Squares Factoring- For all a and b: a^2-b^2=(a+b)(a-b)

Day 2

Notes: aanotes11-3(2).pdf

Justin F.- Discriminant Theorem for Factoring Quadratics: A polynomial ax^2+bx+c can be factored into first degree (linear) polynomials if and only if the discriminant D=b^2-4ac is a perfect square.
Prime/Irreducible: A polynomial that cannot be factored into lower degree polynomials with rational coefficients.
Process to factor polynomials: 1. multiply a and c 2. rewrite b 3. group 1st 2 and last 2 terms 4. take GCF of each set 5. rewrite factors as "stuff" inside and "stuff" outside ( ).

Section 11-4: Estimating Solutions to Polynomials

Notes: aanotes11-4.pdf

This lesson focused on new methods that solve linear and quadratic equations. One of these methods can be done on your graphing calculator, in this method you graph and calculate the zero. The steps are: first you plug the equation into the y='s and graph it. From there you count the number of zeros, or x intercepts, and use 2nd calc, when that screen comes up just go down and go down to zero. With this up on your graph go to one of the zeros and set your boundaries on both sides of the zero, then put in a guess of what you think the zero could equal (note that this guess HAS TO be inside the two boundaries), and repeat this until all of your zeros values have been recorded.
~Austin R.

Shannon S- 11-4 used methods to solve linear and quadratic equations. We learned two methods to use with the graphing calcualtor. Each way begins with graphing your given equation in the y= section. Adjust the window of the graph as needed to see each x-intercept also known as a zero. The first method involves pressing 2nd trace which is the calc function. Press number two, labeled zero. To find the intersections, estimate one value for the right bound and another for the left bound. These values should surround the line. After hitting enter, it will calculate the approximate value of x=0 for that line. Repeat as necessary for the number of zeros you have. The other way is to use the solver. Press the MATH button and hit zero, or scroll until you see the word Solver. Enter your equation above the x= zone, then guess where your line intersects with the x access. Make sure the value you enter is close to the line you wish to solve for. Once again, repeat as necessary.

Section 11-5: The Factor Theorem

Online lesson
Notes: aanotes11-5.pdf

In this lesson we learned that if there are two numbers that are being multiplyed to get the product of zero the last number has to be equal to zero. We also learned two theorems the Zero-Product Theorem for all numbers a and b, ab = 0 IFF a=0 and b=0 and the Factor Theorem which says that x-r is a factor of a polynomial p(x) IFF p(r ) = 0.
There are several names for the zeros
1) Solutions
2) x-intercepts
3) roots
We call numbers zeros when y equals zero
~ Elissa M~

Section 11-6: Factoring Quadratic Trinomials and Related Polynomials

Notes: aanotes11-6.pdf

This lesson talks about factoring polynomials with the quadratic formula, x=-b+ or - the square root of b squared,-4 times a times c all divided by 2 times a. If an equation is too big to factor, or takes too long, you can just use the quadratic formula to factor down the equation. Using the quadratic formula also helps you when the discriminant is not square-rootable, using the quadratic formula can give you an answer. You can also find the zeros using the quadratic formula, but this equation should only be used as a last resort.-Nate F.

Section 11-7: The Rational-Zero Theorem

Notes: aanotes11-7.pdf

Thomas S. - This section is based around the Rational-Zero theorem which deals with the variables ' p ' and ' q '. With these two variables we can find all possible roots of a polynomial by examining the factors of the leading coefficient and final coefficient term(the number w/out the variable).
This theorem is broken down into four easy steps:
1.) Find factors of ' p ' ( Last coefficient , all by itself).
2.) Find factors of ' q ' ( Leading coefficient).
3.) Determine all possibilities for p/q
4.) Then use your calculator and use the possibilities of p/q to help find the zeros

An example: f(m) = 3m³ + 13m² + 21m + 12
Now find ALL possible roots
p = factors of 12: 1± 2± 3± 4± 6± 12±
q = factors of 3: 1± 3±
p/q = 1± 1/3± 2± 2/3± 3± 4± 4/3± 6± 12±
After we input the equation in the calculator we see that the line goes about through the x-axis once around the number -4/3 then if you trace it and input that for x, then x = 0 this means the answer for the zero is -4/3

Section 11-8: Solving All Polynomial Equations

Notes: aanotes11-8.pdf

In this lesson we learned how to find exact solutions to any equation with real coefficients.We also learned how to solve quadratic equations using the quadratic formula. We learned the Fundamental Theorem of Algebra which states that every polynomial equation P(x)= 0 of any degree with complex number coefficients has at least one complex number solutions. There are sometimes things called double roots, which means that the discriminant is O, and the two roots are equal. We also learned the Number of Roots of Polynomial equation Theorem which states that every polynomial equation of degree n has exactly n roots, provided that multiplicities of multiple roots are counted.
Max J. S.
In this section we learned about ways to solve any polynomial equation, regardless of its degree. First we were taught the Fundamental Theorum of Algebra, which says: Every polynomial equation P(x)=0 of any degree with complex coefficients has at least 1 complex number solution. We then learned about double roots, which are two roots of the same value, or when any root appears twice. Along with the double root, we learned about Multiplicity of roots. Multiplicity of roots is the highest power (x-r) of a polynomial when r is a root. All of these definitions lead up to the Number of Roots of a Polynomial Theorum. This theorum says that every polynomial of degree n has exactly n roots, provided that multiplicities of multiple roots are counted. For example, (x-2)^12 would have 12 roots. So basically, in section 11-8, we expanded on ways to solve polynomial equations.

Section 11-9: Finite Differences

Notes: aanotes11-9.pdf

Shannon S and Marisa M: After the in class activity on pages 723-724, we determined that if the first subtraction between the y values of your data then the equation for the set of data will be linear. If the second differences are equal then the equation will be quadratic and if the third differences are equal then the resulting equation will be a cubic. When the differences between the y values are equal, this is the degree. If you have to subtract ten times then the degree of the equation will be 10. If you are given an equation, you must find input and output values in order to find the finite difference. Sometimes, there will be no common difference in which case the equation for that set of data cannot be represented by a polynomial.

Section 11-10: Modeling Data with Polynomials

Notes: aanotes11-10.pdf

In this lesson we learned how to create equations for polynomials. To do this we first have to find the highest degree of that polynomial which is the number of rows it takes to find the same difference. For example, the pattern 1,5,14,30,55, and 91 would be to the degree of 3 because the 3rd row of differences if 2,2,2 and so on. This equation would be in the form y=ax to the 3rd+ bx squared+cx+d. When we have the highest degree we can come up with an equation. To do this by hand (using the example up in the previous sentence) you would want to come up with four equations and then simplify them until you come up with a value for a,b,c and d. The calculator is so much easier though! In your calculator you would enter the data in the edit list under stat. Then you graph the data using zoom 9 (make sure all your other equations are cleared out). Once that is done, go under stat calc and pick the type of equation (we would use Cubic for this equation). Plug that in along with L1 , L2 , Y1. When you click enter you would get your equation
Sarah K.

This lesson expanded on 11-9's idea of rows, but the expansion was into creating equations for the polynomials from the knowledge you get from the difference of rows. To do this by hand you need to first have or create a table of your values to make it easier.