Week 8: Summary of Chapter 3

This week, we had one more lesson to look at till the end of chapter 3, Polynomial and Rational Functions, however we decided to not look at it until a later time due to it being connected to Calculus BC. Since we had a three-day weekend just recently, we only had three days of mathland. We reviewed this week mainly as we had a test on Friday on chapter 3.

To summarize Chapter 3, We began with Polynomial Functions. 

This lesson was simply review of Algebra 2 of End Behavior and Multiplicity. We also looked over even, odd and neither fuctions. 

The most important was multiplicity as it shows how many zeros are there of that one number. Usually this occurs if there is a quadratic function which creates the zero. 

We then looked at Division of Polynomial Functions. 

In this lesson, it is quite obvious we looked at how to divide polynomial functions. To divide you would have the the f(x) and divide it by d(x). To divide it is exactly just as you would divide regular numbers in long division but using x’s and exponents. Below is an example:

Just as shown above the equation’s last exponent on the right side ix x^3. To solve an equation, you would need to put 0’s to fill in the left over pieces. What is needed is shown in red above. You would divide with the number of exponents and number needed to divide. You times the number and put it under the exponent number and then make the numbers negative and you continue till you cannot divide anymore. Another type of division is Synthetic, which is shown below.

You can only divide if the polynomial x has one exponent. ex: x+4

You then take the numbers and plug in any 0’s needed and you divide down and then times the number at the bottom and put that number in the next column and repeat the same process to the end. If the last number is 0 then it is a factor and it has no remainder.

Sometimes you are looking for a remainder and you put the number from the divisor and plug it in for zero. That is the remainder thereom to find the remainder. 

We then looked at Zeros and Factors of Polynomial Functions.

In this one you are looking for the factors and the zeros of polynomial functions. You would divide to find all the zeros. The first polynomial’s exponent tells how many zeros there are. You then find all zeros and tell if they have multiplicity and also the factors. Sometimes the zeros would be imaginary and you would use the quadratic formula to find them. 

After this we looked at Real Zeros of Polynomial Functions.

What would happen if there are no zeros provided to find the rest of the zeros? You use this method which is P/S or Factors of Constant over factors of leading coefficient.

1. You find all the possible zeros from the P/S

2. Test the zeros using remainder thereom

3. Hint- not for test- use graph to see the possible zeros

We then looked at Approximating Zeros.

1. Divide the interval [a,b] in half by finding its midpoint, 

m=a+b/2

2. compute f(m)

3. if f(a) and f(m) have opposite signs, then f has a zero in teh interval [a,m]

if f(m) and f(b) have opposite signs then f has a zero in the interval [m,b] 

if f(m)= o then m is a zero of f. 

You also use the error equation to find out if you are close enough to a zero

error 1/2 (b-a)

the error would need to be  .0005 to be accurate enough to be a zero. 

After this lesson we looked at Rational Functions. 

This lesson was also review from Algebra 2 which consists of vertical asymptotes, horizontal asymptotes and holes. The new thing looked at is slant asymtotes.

To find a vertical asymptote:

set the denominator =0 and find x

To find the horizontal asymptote: 

fx)=anx^n/bmx^m

n=m   y=an/bm

n<m   y=0

n>m   DNE

n= degree of numerator

m= degree of denomator

an= LC of numerator

bm= lc of denominator

also if the HA does not exist (DNE) then there is a slant asymptote. 

To find Slant Asymptote:

Divide the denominator by the numerator and ignore the remainder.

y= answer

Holes:

 Factor the equation and cancel anything possible. What cancels out is a hole, to find the y-int of the hole you plug back in the x to find (x, y)

You would also need to find x-intercepts and y-intercepts:

 Just put 0 for the opposite number to find the answer.

To look more in-depth look at the past posts to understand it better. That is a summary of chapter 3.

See You Next Week!