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Description
Flashcards by Erin Mooney, updated more than 1 year ago
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Created by Erin Mooney
almost 9 years ago
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| Question | Answer |
| Minimums and Maximums of a Quadratic Function | for 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) |
| End Behavior of Basic Polynomials | Determined 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 |
| Using Zeros to Graph Polynomials | If P is a polynomial, then c is a zero (root, x-int) of P if P(c)=0 |
| Intermediate Value Theorem | If 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 |
| Guidelines for Graphing Polynomials | 1. 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 |
| Determining the Shape of a Graph at Zero of Multiplicity m | If 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 |
| Division Algorithm for Polynomials | Polynomial 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)) |
| Remainder Theorem | If P(x) is divided by x-c, then the remainder is the value P(c) |
| Factor Theorem | c is a zero of P(x) if and only if x-c is a factor of P(x) |
| Finding Rational Zeros | 1. 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 |
| Decartes' Rule of Signs | P 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 # |
| Upper Bounds | If 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 |
| Lower Bounds | If 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 |
| Fundamental Theorem of Algebra | Every polynomial with a complex coefficient has at least one complex zero |
| Complex Factorization Theorem | If 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) |
| Zero Theorem | Every polynomial of degree n>=1 has exactly n zeros, provided that a zero of multiplicity k is counted k times |
| Complex/Conjugate Zeros Theorem | If 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) |
| Rational Functions | f(x)=p(x)/q(x) where p(x) and q(x) are polynomials |
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