Lesson 7.3-7.4 Video

Lesson 7.5 video

Lesson 7.1 Video

Journal Write #4

*Fundamental Theorem of Algebra*
In this theorem, every polynomial with a degree greater than 0, has at least one root in the set of complex numbers. For example, if you had the roots -2,-i, i, you could factor and solve the problem.

*Complex Roots to Polynomial Equations*
When you have your two roots, you have to factor it. Then you simplify and solve like in this example shown below.

*The Conjugate Pair Theorem*
In this theorem, if a±bi is a zero of f(x), then a∓bi is also a zero. Like in the video I watched, suppose that has these were zeros in the picture shown below. You would have to find 3 more zeros like in this example.

Journal Write #5-Precal

*Introduction to Rational Functions*
A rational function can be written as f(x)=(2x^2-5x-3)/(2x-5). In this equation is 5/2. You always have to have your numerator equal to 0. Then factor your equation and your zeros are x=-1/2 and x=3.

*Asymptotes of a Rational Function*
To find the vertical asymptote, I have to factor out the numerator of the problem and simplify. In this given equation (5-x^2)/(x+3), its vertical asymptote equals x=-3.

For the horizontal asymptote (5-x^2)/(x+3), since the degree in the numerator is different from the degree in the denominator, it means that there are no horizontal asymptotes.

In this case since the degree in the numerator is one number larger than the degree in the denominator, there will be an oblique asymptote. To find this out you have to use long division.

*Graphing a Rational Function*
For graphing a rational function I had to find the x-integer y-integer, vertical asymptote, and horizontal asymptote. It was a bit hard to understand.

Journal Write #5-Al2

For basic exponent properties when you when multiplying your question might look similar to a^m*a^n=a^(m+n). I understand that I have to multiply the numbers in front of the exponent and add the exponents together to get your answer. When you divide you will will have a different property like a^m/a^n=a^(m-n) which means you divide both numbers in front of the exponents, and then subtract the exponents to get your answer. There are many properties.

In the next video I learned about negative exponents for example 5^-1/1. So first I have to switch 5^-1/1and make it positive just as shown in the bottom picture. Then you multiply and divide and get a fraction of 1/5. There is also another example under it.

In the third video for complex numbers I understood a little bit of how to do it. I learned that complex numbers are in the form of z=a+bi. A is the real number and the B is imaginary. In the video I was given five complex numbers to plug into separate equations. I was given z1+z2, z1-z2, z1*z2, and z1*z5.

Journal#4-Solving Quadratic Equations

An example of this type of equation is in this picture shown below.

Step 1) See which two numbers multiply to get 6 and that add up to your middle term 5.

Step 2) Separate your x^2 and add an x inside each parenthesis.

Step 3) Now add in the numbers you multiplied to get 6.

Step 4) Don’t forget to add a zero at the end of the equation.

Here is another example called completing the square, which is still based on solving quadratic equations.Step 1) That first piece is your coefficient and you want the x^2 to equal to 1.

Step 2) Separate the equation as shown below.

Step 3) Take the number that’s in front of the x term and multiply. Once you get your answer, square it, and place that answer in the blank area in the top of it.

Step 4) Subtract 9 on the other blank sign then begin to simplify.

Step 5) Now you square root on both sides of the equation, and keep the plus minus sign on the 1.

Step 6) To finish up, you will have two equations to solve. On one side you’re solving with your positive 1, and on the other you solve it with a negative 1, and now you have 2 solutions.

*Descartes’ Rule of Signs*

In this given example h(x)= 2x^4-3x+2, you have to see how many times the sign changes. I this case there are 2 which means that there could be 2 or 0 positive real numbers. To test it out we have to make all the x’s negative.

Remember when there is an even power your sign will go away, and if there is an odd power the sign will be in front of your number.

Now that we reduced the problem, there were not any sign changes which means that there are no negative real zeros. Here is another example. At the end you had to use the rational zero test and then simplify. If you don’t know how to use the rational zero test go to my link so you could learn it https://ericasgeo.wordpress.com/2013/01/14/rational-zero-test/

*Rational Zero Test*

I learned in this example which numbers to factor. Then I was suppose to divide p with all the potential denominators of q. Then I simplify those numbers which become potential rational divisors.

*Synthetic Division*

This is what I learned. For the problem x^2+2x-8/ x-2 you have to see how you can make your denominator a 0. Step 1) Create an upside down bracket and place a 2 outside as shown in the picture. Also placing your coefficients inside the bracket going across from the 2.

Step 2) Drop down 1.

Step 3) Multiply 1&2 and place answer underneath the 2 inside the bracket.

Step 4) Add 2 and 2 and place your answer below. Then again, multiply 4&2 and place answer underneath -8 inside the bracket.

Step 5) Subtract both -8 and 8 which is 0, and 0 is your remainder.

Step 6) So your 1(which is x) and your 4 (which is your constant term) will drop down and x+4 will be your answer.