Math Ch. 2 Test

If the zeros of a function are -1, 2, and 3+i, how do you write a function?

(x+1)(x-2)(x-(3+i))(x-(3-i)) then multiply it all out.

How do you determine the max # of turning points?

n-1

How do i find the x-intercepts?

replace y with 0 and solve for x

What does the p represent?

the constant (the number at the end with no variable)

How do you find out if the graph will ever cross the asymptotes?

whatever the asymptote is, make that = to the given function and and find if it is true. ex. 0≠-3 so no it will not 0=0 so yes it will cross the asymptote

What is the factor theorem?

x-r is a factor of polynomial P(x) iff R(x)=0.

How do you find the vertex of a function?

x=-b/2a (vertex formula) -This gives you the x of the vertex, then plug it back into the function to find the y.←

How do you find the horizontal asymptotes?

x→-∞, f→ x→∞, f→ The end behavior represents the horizontal asymptotes. If the end behavior is ∞, then there is no horizontal asymptote.

What is vertex form?

y=(x-h)²+k

If a polynomial is being divided by x+1, what would you put in the box on synthetic division?

-1

What are the rules for the discriminant on determining the number of zeros in a function?

-b²-4ac >0, f has 2 real zeros -b²-4ac =0, f has 1 real 0 multiplicity 2 -b²-4ac <0, f has 2 complex conjugate zeros (not real)

When asked to find the zeros of an equation like P(x)=10x³-17x²-7x+2, what do you do?

1. Find the possibilities for the p and q 2. Put all ps over all qs 3. The the p/qs in the equation to see which one equals 0 4 Do synthetic division with the one found and simplify from then on.

What do you need to find when graphing Rational Functions?

1. Vertical Asymptotes (and holes) 2. Horizontal Asymptotes 3. Slant (oblique) Asymptote 4. X & Y intercepts 5. Additional Points Needed 6. State D & R

What is the Fundamental Theorem of Algebra (2 parts)?

1. polynomial FUNCTION of degree n>0, has at least ONE ZERO in the COMPLEX numbers. 2. Every polynomial EQUATION of degree n>0, has at least one root (solution) in the complex numbers.

What is the corollary to the fundamental theorem of algebra?

A polynomial function of degree n>0, has exactly n zeros including multiplicities in the complex numbers.

What is a rational function?

A ratio of 2 polynomial functions. Ex. R(x)=f(x)/g(x), g(x)≠0 Dr = Df ∩ Dg except x where g(x)=0

If a number is divided by a very small number, what will the result be?

A really big number, ∞

What types of zeros come in pairs?

COMPLEX ZEROS (imaginary zeros)

How do you put into vertex form?

Complete the square

W(x) =3x³+x²-5 Degree? LC? a₀?

Degree=3 LC=3 a₀=-5

How do you solve polynomial inequalities?

Factor them and place the solutions on a number line. -If it is ≥ or ≤ then filled in circle unless in denominator -If it is < or > then open circle Then find where it is positive and negative on the graph State the solution

What are the different ways to find the x-intercept?

Factoring Complete the Square Quadratic Formula

How do you solve rational inequalities?

First simplify and get the inequality = to 0 FInd the critical points -make the numerator and denominator = to 0 and solve to find the critical points. Graph and find where the graph is positive and negative State where the solution is on graph.

What is Descartes Rule of Signs?

If P(x) is a polynomial function in descending powers, then 1. The number of sign changes in P(x) gives the maximum number of positive zeros, and 2. The number of sign changes in P(-x) gives the maximum number of negative zeros. *when find P(-x) only the odd degree values change*

What is the Intermediate Value Theorem?

If a function is continuous between a and b, then the function takes on every value between f(a) and f(b).

What is the Complex COnjugate Theorem?

If a+bi is a zero of a quadratic function, then a-bi is also a zero. *Complex zeros come in pairs*

What is the Remainder Theorem?

If polynomial, P(x), is divided by x-r, the remainder in a constant P(r). Ex. f(x)=x⁵+8x³+2x-15 at x=2 with 2 in the shelf divided by x⁵+8x³+2x-15, the remainder is 85 and that is equal to the f(2)=85

How do you know when there is a slant asymptote?

If the degree of the numerator is exactly one more than the denominator.

What does the intermediate value theorem help do?

It helps locate real zeros of polynomial function when given a continuous function.

Explain this using the Intermediate Value theorem: Temperature 75 degrees at 8AM and 90 degrees at 3 PM.

It started at 75 degrees and could've went down then up and possibly above 90 then back down to 90, but it had to take on the values 75-90.

How do you find the vertical asymptotes?

Make g(x)=0 then solve and find the vertical asymptotes. Holes are possible.

What is significant about V(x)=0?

No a polynomial because aₙ≠0

Define a polynomial function.

P(x)= aₙxⁿ+aₙ−₁xⁿ⁻1+aₙ−₂xⁿ⁻2+.....+a₂x²+a₁x+a₀, where aₙ, aₙ−₁, aₙ−₂,.....a₂, a₁, a₀ ∈ all real numbers, aₙ≠0 and n∈W n is the degree of the polynomial aₙ is the leading coefficient a₀ is a constant

Know how to do long division.

STUDY

Know how to do synthetic division and to make sure to add in 0s when skipping a value.

STUDY

What does the q represent?

The value of the leading coefficient (number in front of variable with highest power)

If given a continuous function f(x) and f(1)=-2 and f(2)=3, what do we know about this?

Then ∃ (there exists) x such that f(x) =0 (or -1, -2, -1.768686, or any value between -2 and 3) by IVT. *We don't know the value of X but we know that it exists*

How do you find the slant asymptote?

Whatever the problem is, if the degree of numerator is exactly one more than the degree of the denominator, then perform long division and the polynomial you get out of that is the equation for the slant asymptote.

How do you know if there is a hole?

When simplifying the original polynomial, if there is a restriction after simplifying, then that restriction is a hole. Plug that restricted value into the simplified polynomial to find the y value.

What is the discriminant and what does it do?

b²-4ac (part under the square root in quadratic formula) -It tells us how many zeros there are

What is the standard form of a quadratic function?

f(x)=a(x-h)²+k a≠0 (h,k)=vertex (when writing equation, use point and vertex to find a, then use that for equation)

What is the definition of a quadratic function?

f(x)=ax²+bx+c a, b, c ∈ real numbers a≠0