Chapter 11: Polynomials

Chapter 11 Preview: aach11preview.pdf
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Wiki Summary Assignments

11-1: Abigail B. and Isaiah C.
11-2: Curtis K. and Abigail B.
11-3: Brittani H. and Tricia L.
11-4: Jacob R. and Amanda T.
11-5: Nate. J. and Jacob R.
11-6: Josh H. and Jessica R.
11-7: Andrew E. and Tricia L.
11-8: Jeremiah B. and Tyler Y.
11-9: Amira A. and Abigail B.
11-10: Courtney G. and Brittani H.

Section 11-1: Introduction to Polynomials

Notes: aanotes11-1.pdf

Lesson 11-1 is all about introducing polynomials. We learned about the basics like degrees, terms, and coefficients. The degree is the highest exponent in the polynomial, a term is [the number before each variable], and the leading coefficient is the number that goes along with the highest degree exponent. There are a few different types of polynomials. Degree 1 is linear, degree 2 is quadratic, degree 3 is cubic, and degree 4 is quartic. We can also use polynomial functions to solve real-world problems using polynomials. -Abigail B. and Isaiah C.

Section 11-2: Polynomials and Geometry

Notes: aanotes11-2.pdf

This lesson applies polynomials to geometry. We learned to classify certain polynomials by the number of terms they contain. A monomial has one term, a binomial has two terms, and a trinomial has three. If the polynomial has more than three terms there is no specific name given to it. We also learned to find the degree of a polynomial. It is the largest sum of the exponents of the variables in any term. To find the degree of 4x^3 + 2n^4, you would add 4 and 3 to get a degree of 7. When multiplying polynomials (excluding monomials) you need to use what's called extended distribution. In other words, multiply every term in one polynomial by every term in the other. - Curt K

Lesson 11-2 is about applying polynomials to geometry. We learned about monomials (one term), binomials (two terms), trinomials (three terms), and anything with more than three terms is a polynomial. When you have a polynomial with more than one type of variable, you take the sum of the exponents of the variables in the terms. When you have a problem like (2x + 5)(10x³ - 2), you have to used extended distribution. This just means that you multiply each term in the first polynomial by each term in the second polynomial. - Abigail B.

Section 11-3: Factoring Special Cases

Day 1

Notes: aanotes11-3(1).pdf

In this lesson we learned how to factor which is rewriting a polynomial as a product of factors. There are three types of factoring: greatest common factor, binomial square factoring, and the difference of squares factoring. Binomial square factoring is when you have a perfect-square trinomial and the first and last term are square roots. The middle term has to be the product of the first and last terms square and doubled for it to be factorable. Binomial square factoring is for all real numbers a and b: a^2+2ab+b^2=(a+b)^2 or a^2-2ab+b^2=(a-b)^2. The difference of squares factoring is when you just take the square root of both of the terms and put them into two expressions with the one being addition and the other one subtraction. The difference of squares factoring is for all real numbers a and b where a^2-b^2=(a+b)(a-b).
Brittani H.

Day 2

Notes: aanotes11-3(2).pdf

For Chapter 11 Lesson 3 we took two days to learn all of the different ways to factor expressions depending on the form that they are written in. The first thing you need to now is how to factor which is when a polynomial is written as the product of factors that multiply to make that number. The first way to factor is by using the Greatest Common Factor which is when you find number and variables that are divisible to all part of the expression and leave that outside the parentheses with the new expression inside. The next way is by binomial square factoring which is when you have a perfect-square trinomial and the first and last term are square roots. The middle term has to be the product of the first and last terms square and doubled for it to be factorable. The other way to factor is using the difference of squares factoring which obviously has to be a subtraction expression. When you do this you take the square root of both of the terms and put them both into two expression with one addition and one subtraction. In day two of learning we took on the discriminant theorem for factoring quadratics which means that if the discriminant(b²-4ac) is a perfect square then it is factorable and you can proceed on with the problem. Any polynomial that can't be factored to a lower-degree with rational coefficients means that it is prime.(irreducible) Then we learned how to factor this down in 5 steps and later on reducing to three easy steps. The first thing you want to do is to multiply a and c and then you want to factor ac so that the factors add up to b. When you find the factors you then replace them with b and group the first two and last two terms of the expression. Find the greatest common factor and rewrite the "stuff" inside and the "stuff" outside in parentheses repectively and that covers all the basic essential to lesson 3.
-Tricia L.

Section 11-4: Estimating Solutions to Polynomial Equations

Notes: aanotes11-4.pdf

In is lesson we learned how to find and identify the x-intercepts of a graph. This is done is by graphing the polynomial on a calculator and finding the zeros (places where the graph crosses the x-axis). It is possible for a graph of a polynomial not have any x-intercepts. The number of possible x-intercepts a polynomial can have is equal to the degree of the polynomial. In addition to graphing the polynomial to find the intercepts many calculators have a solving feature, which can give you a more precise answer. -Jake R.

Section 11-5: The Factor Theorem

Online lesson
Notes: aanotes11-5.pdf

Lesson 11-5 is based on The Factor Theorem. The lesson starts with stating that a product of two numbers equals zero iff one of the factors equals zero. The Zero product theorem immediately followed. It says that for all a and b, ab equals zero iff a equals zero or b equals zero. This is only true when a and b are expressions. Another theorem is the x minus r is a factor of a polynomial P(x) if and only if P(r) equals zero. Also in this lesson we worked on finding zeros by factoring. We also factored by finding the zeros. We then can find equations from the zeros. -NJ

Section 11-6: Factoring Quadratic Trinomials and Other Related Polynomials

Notes: aanotes11-6.pdf

In lesson 11-6 we learned how to factor quadratic trinomials and related polynomials. To do this we need the quadratic formula which is x= -b +- square root of b squared - 4 ac all over 2a. You will use this when you have a very large coefficient. ANY quadratic polynomial can be factored by using the Quadratic Formula. Factoring polynomials of a degree n > 3 is easy. You need to find a common monomial. After you do that you use the Zero Product Theorem. Then the last step you use the Quadratic Formula to solve the quadratic equation.
- Josh H

Section 11-7: The Rational-Zero Theorem

Notes: aanotes11-7.pdf

In Chapter 11 Lesson 7 we learned the rational-zero theorem which in short terms means that we can find any and all possible roots of a polynomial. To do this we examine the leading coefficient and the final term without a variable. There are four basic steps to use the rational-zero theorem and the first one is to find the factors of "p" which is the last coefficient down at the end all alone. The second thing is to find the factors of "q" which is the leading coefficient. The next thing you want to do is determine all the possible roots by dividing p/q. Then you use your calculator to graph the equation and you use the possible roots to guess about which ones are correctly shown on the graph at (x,0). Here is an example that will help you understand this concept:
f(x) = 4x⁴+3x³-2x-24. First we factor the lone coefficient which in this case is -24. So p=factors of -24: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. The leading coefficient would be the number which is before the highest exponent which is 4. Then we factor for q=factors of 4: ±1, ±2, ±4. -Tricia

In lesson 7 of chapter 11 we learned what was the Rational-Zero Theorem. This theorem suppose that all the coefficients of the polynomial function, and the integers with a cant equal zero and a can't equal zero. Then p over q is a zero of f, then p is a factor of a0 and q is a factor of an. In an automatic grapher it can't estimate zeros of the function which are highly accurate. When zeros are rational numbers it's possible to get an exact answer using the Rational-Zero Theorem. Andrew E.

Section 11-8: Solving all Polynomial Equations

Notes: aanotes11-8.pdf

We learned in lesson 11-8 about solving any polynomial equation. To help us do that we went over some theorems. One was the Fundamental Theorem of Algebra. Another was The Number of Roots of a Polynomial Equation Theorem. We also learned about a Double Root and a Multiplicity of a Root. You can either solve for the zeros of the equation by factoring down then setting it to 0 and solving. You can also use the quadratic formula. Or you can find P then find Q then put P/Q and see which one/ones work.
Tyler Y.

In Lesson 11-8 we learned how to solve all polynomial Equations. There are several theorems that will help solve these equations. The first theorem is the Fundamental Theorem of Algebra. This theorem says for every polynomial equation P(x)= 0 of any degree with complex coefficients at least one complex number solution. We also defined what a Double Root is and Multiplicity of a Root. A double root is in a quadratic, when the discriminant equals 0, there will be two roots that have the same value (when any root appears twice). Multiplicity of a Root is the highest power of (x-r) of a polynomial when r is a root. The next theorem is the Number of Roots of a Polynomial Equation Theorem which states that every polynomial of a degree "n" has exactly "n" roots (including multiplicity). *You can find out how many roots an equation has by its degree. (Ex: x^15+1=0, degree is 15). By finding the roots we can find out how many possible real solutions there are.
Jeremiah B.

Section 11-9: Finite Differences

Notes: aanotes11-9.pdf

11-9 was pretty much just an introduction to what we learned in 11-10. We started with data in a table, and found the differences of the y-values. The number of times that we had to find the differences until they became equal was the degree of the polynomial. First differences are linear, second differences are quadratic, third differences are cubic, and fourth differences are quartic. The Polynomial-Difference Theorem says that however many times you have to take the difference, that is the degree of the polynomial equation. This info helps us later on find the actual equation from the data. -Abigail B.

Section 11-10: Modeling Data with Polynomials

Notes: aanotes11-10.pdf

In lesson 11-10 we learned how to find an equation from a informational data table. To find the equation you substitute three known order pairs from the table into the general equation y=ax^2+bx+c. These three pairs put into the equation has produced a system of three equations, and to solve this system, reorder the equation so that the largest coefficients are on the top line. Then subtract each equation from the one immediately above it. Once you find what each variable is you can plug it into the equation y=ax^2+bx+c. You can also find the equation for the data by plugging in the values and solving it with your calculator.
Brittani H.