A polynomial function only has whole/positive numbers in exponents

Leading Coefficient:
The term with the highest degree

End Behavior:
As x -> -∞, f(x) -> something
As x -> +∞, f(x) -> something

If the leading coefficient is positive f(x) goes up as x goes up, and if it’s negative it goes down

If the leading coefficient is even then both sides go the same way, and if negative the one way goes down and one way goes up

Long Division:

When you divide by something that isn’t x – k

Synthetic Division:
When the thing you divide by x – k
You can also find the y-value if you put in an x-value
You put in k from x – k (remember if it’s + there is a – in front)
Use 0 for any missing terms, the top part is the number in front of every term as the power gets smaller but if there is f.eks. Isn’t an x^2 value you wouldn’t skip it, but actually put a 0 there

All possible rational roots / Rational Root Theorem:
p/q
p: The constant part of the equation (no x)
q: The number before the x with the highest power
Ex.
x^3 + 2x^2 – 4x + 20
p: 20
q: 1

The Remainder Theorem:
When using synthetic division if the remainder is 0, then the number you divided by is a solution to that equation

You can also find the zeros of a function with a higher power than 2, by using synthetic division
And when you find one of the solutions through the remainder theorem, you get a new equation and then you can factor that and then you have the answer

The polynomial you get out of synthetic division is called a depressed polynomial

Sum or difference of cubes:

How to solve a polynomial function:
Make it into a quadratic function using an auxiliary variable (z)

Polynomial long division to find zeroes:

P(x) = d(x)    *    Q(x)     +     R(x)

Divisor  –  Quotient  –  Remainder

This all just means using long division to simplify the function down from a 4th, 5th, 3rd degree down to a quadratic function which can be easily factored
So the composition above is just the outcome of a long division, where you get a function and a remainder (Nothing special)

1. You find one zero by just guessing/trying to plugin etc.
2. Use long division or synthetic of that zero in the x-k form and divide by the bigger function
3. And then you should have a quadratic function which you can then factor out, and then you have all of the zeroes

Smart tricks:

1. The constant in a polynomial is the product of all of the solutions/zeroes
2. Cubic functions always have at least 1 x-intercept
3. If one of the factors gives is a 2nd degree then the point is a bounce and it doesn’t go through that zero

Remember a function that has to do with years:
Remember that usually “x” is years after the first year and not the year itself
Ex.
2005 would be 14 if the first year was 1991

Rational Zero Theorem:
You always have to remove complexity or irrationality from polynomial zeroes

Descartes Rule of Signs:
You can see how many positive and negative zeroes are in a polynomial

1. Find the positivity or negativity of each term and write it down
2. The number of changes will equal the number of positive zeroes
3. Now substitute x with -x and then check how many changes you get, and that will be the number of negative zeroes

Ex.

How to find zeroes in a polynomial with rational terms if you know some zeroes:
You don’t have to calculate new zeroes if the zeroes you know are complex or irrational
Ex.
x1 = 4
x2 = sqrt(3)
x3 = 7 – 2i
Then you can find x4 and x5, by simply taking the conjugates
x4 = -sqrt(3)
x5 = 7 + 2i
This happens because those conjugates will cancel out each other

Multiplicity:
If you factor a function and one linear factor is there more than once it is called multiplicity, and the amount is the same as the exponent
Ex.
(x+5)^3 * (x-1)
x = -5, 1
-5 has a multiplicity of 3

Turning points:
n-1, where n is the highest exponent
Ex.
x^6 + 3x has 5 turning points

The intermediate Value Theorem:
You can find out if a function has a zero between two points f(a) and f(b).
If f(a) and f(b) have different signs AKA one is positive and the other one is negative, then there is a zero between them.