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)

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

**Smart tricks:**

- The constant in a polynomial is the product of all of the solutions/zeroes
- Cubic functions always have at least 1 x-intercept
- 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

- Find the positivity or negativity of each term and write it down
- The number of changes will equal the number of positive zeroes
- 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.

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