Flashcards
Chapter 4
Cards in this deck (18)
- 1Minimums and Maximums of a Quadratic Functionfor quadratic function f(x)=a(x-h)^2+k, the min/max value occurs at x=h
-if a>0, f has min. value of k at x=h
-if a<0, f has max. value of k at x=h
-min/max is usually f(-b/2a) - 2End Behavior of Basic PolynomialsDetermined by degree n and sign of leading coefficient a
-p(x) with even degree: if a>0, is upward parabola
if a<0, is downward parabola
-p(x) with odd degree: if a>0, is snakey from bottom left to top right
if a<0, is snakey from top left to bottom right - 3Using Zeros to Graph PolynomialsIf P is a polynomial, then c is a zero (root, x-int) of P if P(c)=0
- 4Intermediate Value TheoremIf P is a polynomial and P(a) and P(b) are opposite signs, then there is at least one c between a and b such that P(c)=0
- 5Guidelines for Graphing Polynomials1. find zeros, factor
2. test points, make table of values. determine if graph is above or below x-axis
3. determine end behavior
4. graph - 6Determining the Shape of a Graph at Zero of Multiplicity mIf c is a zero of P(x) and its corresponding factor (x-c) occurs exactly m times in the factorization of P, then c is a zero of multiplicity m
- 7Division Algorithm for PolynomialsPolynomial P(x) (dividend) and D(x) (divisor) and unique polynomials R(x) (remainder) and Q(x) (quotient):
P(x)/D(x)=Q(x)+(R(x)/D(x)) - 8Remainder TheoremIf P(x) is divided by x-c, then the remainder is the value P(c)
- 9Factor Theoremc is a zero of P(x) if and only if x-c is a factor of P(x)
- 10Finding Rational Zeros1. list all rational possible zeros
2. use synthetic division and test the possible rational zeros. record quotient when found
3. repeat process with recorded quotient - 11Decartes' Rule of SignsP is a polynomial with a real coefficient
1. the # of + real zeros of P(x) is either equal to the # of variations in sign in P(x) or is < that by an even #
2. the # of - real zeros of P(x) is either equal to the # of variations in sign in P(-x) or is < that by an even # - 12Upper BoundsIf we divide P(x) by x-b (with b>0) using synthetic division and if the row containing the quotient and remainder >=0, then b is an upper bound for the real zeros of P
- 13Lower BoundsIf we divide P(x) by x-a (with a<0) using synthetic division and if the bottom alternates in sign, then a is a lower bound for the real zeros of P
0 can be positive or negative - 14Fundamental Theorem of AlgebraEvery polynomial with a complex coefficient has at least one complex zero
- 15Complex Factorization TheoremIf P(x) is a polynomial of degree n>=1, then there exists complex numbers a, c1, c2, c3,..., cn such that P(x) can be factored into P(x)=a(x-c1)(x-c2)...(x-cn)
- 16Zero TheoremEvery polynomial of degree n>=1 has exactly n zeros, provided that a zero of multiplicity k is counted k times
- 17Complex/Conjugate Zeros TheoremIf a polynomial P(x) has real coefficients and if the complex number z=a+bi, then the conjugate of z, z=a-bi is also a zero of P(x)
- 18Rational Functionsf(x)=p(x)/q(x) where p(x) and q(x) are polynomials
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- College Math 110 Flashcards on Chapter 4, created by Erin Mooney on 06/07/2015.
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- Erin Mooney
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