POLYNOMIAL FUNCTIONS

What can the degree distinguish for the polynomial?

If even, the polynomial arm directions are the same. ; If not, the polynomial arm directions are not.

What can the leading co-efficient distinguish for the polynomial?

If positive, that means the right arm goes upwards. ; If negative, that means the right arm goes downwards.

How to tell if (x-1) is a factor

If the co-efficients add up to equal zero

What does setting the equation to zero mean

The net means to solve the roots are independent of the initial graph and final solution. (Like Hess's Law)

Zero interactions in polynomial form

1 multiplicity = goes through; even multiplicity = tangent ; odd multiplicity = inflection point

Unique Factorization Form

1. Assumes that all polynomial equations can be written with a leading co-efficient and n amount of linear factors where a represents the zeros 2. Every polynomial will have the same number of factors as degree.

Factor Theorem

A polynomial f(x) has a factor x - k if and only if f(k) = 0

Why is it impossible to tell the degree given a graph of a polynomial function

Because tangent and inflection points only require even or odd, thereby making it impossible to distinguish the exact combination of multiplicities.

How to calculate leading co-efficient, degree and constant in any polynomial multiplied by any polynomial?

Constant and LC is separated by highest and lowest degree and (assuming polynomial form), is therefore just the first-first and last-last.

What to do before factoring a polynomial?

Convert to polynomial form.

Polynomial Complexity

GCF and polynomial form

What redundancy does synthetic division get rid of?

Having to write down multiplying divisor by exactly how much of quotient is.

What does slope denote?

The interval it increases by

What type of multiplicity can you instantly get a factor from?

If the graph crosses the x-axis, it is guaranteed (x-a)

When to use unique factorization theorem and when to use rational zero theorem

If you are given a polynomial function with a point, use unique factorization. Otherwise, use rational zero if you have a graphing calculator.

Remainder Theorem

If you divide a polynomial P(x) of degree n > 1 by x - a, then the remainder is P(a).

Three forms of polynomial functions

Integral, rational, real

How does factoring a trinomial work?

It is like a system of equations. In the nature of FOIL, the outer terms and inner terms are multiplied by the same numbers. However the middle term then adds up these numbers. That is why we need a number that adds up to equal the middle term given but multiplies to equal what the leading co-efficient and constant equal.

What does the constant say about the y-intercept?

It is the y-intercept.

LDMBS (Lick Da Mule's Ball Sack)

Leading co-efficient, Degree, Multiplicities, Bends depend on m, symmetry

Difference between multiplicity, number of zeros and degree

Multiplicity = amount of times a zero repeats; Number of zeros = distinct zeros; Degree = total multiplicities

An error you made calculating the constant of the polynomial function?

One of the zeros was literally zero. This means the other constants are zero. Another time, a zero had a multiplicity of three and you forgot to cube it before multiplying by the other last numbers in the factor.

What indicates an unreal solution to a polynomial graph?

Spontaneous deviation

Why can't a symmetrical graph of two tangents upwards not represent a polynomial function of degree 10?

Symmetricalness implies that the multiplicities are the same (5). However tangent shape implies it must be even. This contradicts.

As long as a polynomial function is in polynomial form...

Synthetic division and regular division works since leading co-efficient and constant come from one multiplication

What does synthetic division fundamentally depend on?

That the co-efficient of the x in the divisor (x-a) is one.

Integral Zero Theorem

The idea that if a = 1, the zero is a factor of the constant.

Rational Zero Theorem

The idea that if a > 1, the zero is in the form p/q where p is a factor of the constant and q is a factor of the leading-coefficient.

What do non-zeros always come as?

Two non-real distinct solutions with one multiplicity each

What kind of a slope is undefined

Vertical

Polynomial form

Where A is a real number and x goes the power of one or greater

Can you convert a non-real zero polynomial into unique factorization form?

Yes. It would be impossible to graph but a would be where you put the impossible zero.

An error you made that was really close to the right answer

You forgot to take into account the leading co-efficient when you were stating the negativity of the constant (last number of factors and a value)