Ch 3 Honors Precalc

What is a polynomial

A function that is always smooth and continuous (no holes, breaks, or sharp points)

root multiplicity

• mult = 1: straight through

• mult = even: bounce

• mult = odd: wiggle

degree and leading coeff. effect

• odd degree, positive leading coeff:

  • as x → infinity, y → infinity

  • as x → - infinity, y → -infinity

• odd degree, negative leading coeff:

  • as x → infinity, y → - infinity

  • as x → - infinity, y → infinity

• even degree, positive leading coeff:

  • as x → infinity, y → infinity

  • as x → - infinity, y → infinity

• even degree, negative leading coeff:

  • as x → infinity, y → - infinity

  • as x → - infinity, y → -infinity

local extrema for a polynomial with degree n

n - 1

Intermediate Value Theorem

If P is a polynomial function and P(a) and P(b) have opposite signs, there exists at least one value c between a and b for which P( c) = 0

Division Algorithm

h(x) = f(x)/g(x) = q(x) + r(x)/g(x)

• q(x)  = quotient, end behavior

• r(x) = remainder, if you set r(x) to 0 and solve for x (as long as r(x) has the variable x) you will get the intersection point of r(x) and q(x)

• If the degree of f(x) < g(x), there exists a H.A. y = 0

Rational Root Theorem

for a function with leading coeff. a and constant b, the possible rational roots will be +- factors of b/factors of a

Remainder theorem

If the polynomial P(x) is divided by (x - c), the remainder is P( c)

End Behavior notation

as x → a⁺ means from right → left

as x → a^- means from left → right

Solve polynomial/rational inequalities

My - Move all terms to one side

Friendly - Factor the polynomial

Ferret - Find the cut points

Makes - Make a sign chart

Sandwiches - Solve