Rational and Polynomial Relationships

Review

Use the following vocabulary to describe each Expression, equation, term, factor, coefficient, variable, zero, function, domain, range F(x) = 3(x + 2)(2x + 5)( 𝑥 2 + 4)

Preform the following operations

Polynomial Division • Write 𝑎(𝑥) 𝑏(𝑥) as q x + 𝑟 𝑥 𝑏 𝑥 • Divide

• If Remainder Theorem

p(x)

/

(x – a)

= q(x) with remainder r(x), then p(x) = (x – a) q(x) + r(x) • Example:

(x^3 – 7x – 6)

/

(x – 4)

= x

2

+ 4x + 9 with remainder 30, So… x

3

– 7x – 6 = (x – 4) (x

2

+ 4x + 9) + 30.

• Divide the following writing the answer in terms of the remainder theorem 𝒙 𝟑 +𝟑𝒙 𝟐 +𝟓𝟓 𝒙 −𝟕

Re-write using remainder thrm.

POLYNOMIAL GRAPHING TECHNIQUES AND FACTORIZATION

Strategies for visualizing polynomial graphs • Input / Output Table • End Behavior - even and odd degree functions • • Y intercept Descartes Sign change • Factoring / find zeroes • Remainder Theorem • Rational Zero Theorem (p/q) • Quadratic Techniques • Relative Minimums and Maximums by apprx.

• • • • Quick Sketch using end behavior A positive quartic function A negative quartic function A positive cubic function A negative cubic function

Explain the Fundamental Thrm. of Algebra

Descartes Sign Rule The sign changes in f(x) gives the number of positive zeroes or an even increment of zeroes below that number The sign changes in f(-x) gives the number of negative zeroes or an even increment of zeroes below that number

Determine number of positive, negative, imaginary zeroes

Write and sketch a polynomial function given the roots

Write and sketch a polynomial function given the roots

Given the function and a root determine other roots

Factor

• If Remainder Theorem

p(x)

/

(x – a)

= q(x) with remainder r(x), then p(x) = (x – a) q(x) + r(x) • Example:

(x^3 – 7x – 6)

/

(x – 4)

= x

2

+ 4x + 9 with remainder 30, So… x

3

– 7x – 6 = (x – 4) (x

2

+ 4x + 9) + 30.

• When is the remainder theorem a useful tool?

Rational Zero Theorem (p/q) • Determine all possible rational zeroes for the following polynomial function

Quadratic Techniques • Factor the following polynomial

Factor and graph the following polynomials

Squareroot functions • Graph the following squareroot functions

RATIONAL EXPRESSIONS AND FUNCTIONS

Preform the following operations

Strategies for visualizing rational graphs • • • • Input / Output Table Transformations of the parent function (1/x) Holes and Asymptotes Y intercept and X intercept(s)

Horizontal and vertical asymptote rules

If n < m , then the x axis is the horizontal asymptote If n= m , then the line y = a/b is the horizontal asymptote If n > m , then there is no horizontal, it is instead a slant or oblique if n is greater than m by one degree then the quotient of the function is the slant asymptotes

Determine any holes or asymptotes Why are some excluded values holes and others vertical asymptotes?

Graph