Math Vocab Chapter 2

solution; (x-a); x-intercept

If x+a is a zero of a polynomial f, then the following three statements are true: A. x=a is a ------ of the polynomial equation f(x)=0 B. ----- is a factor of the polynomial f(x) C. (a,0) is an ------------- of the graph f

complex conjugates

the numbers a+bi & a-bi are called ----------------- & their product is a real number a^2 + b^2

improper; proper

the rational expression p(x)/q(x) is called ------ if the degree of the numerator is greater than or equal to that of the denominator, & is called ----- if the degree of the numerator is less than that of the denominator

n; n-1

a polynomial function of degree n has at most ----- real zeros & at most -------- turning points

intermediate value theorem

the ------------------- states that if f is a polynomial function such that f(a) does not equal f(b), then in the interval [a,b] f takes on every value between f(a) & f(b)

zeros; undefined values

the critical numbers of a rational expression are its ----- & its ------------

P=R-C

the formula that relates cost, revenue, & profit is ---------

axis

the graph of a quadratic function is symmetric about its -------

continuous

the graph of all polynomial functions are ----------, which means that the graphs have no breaks, holes, or gaps.

i = square root of -1; i^2 = -1

the imaginary unit i is defined as ----, when ----------

rational zero test

the test that gives a list of the possible rational zeros of a polynomial function is called the -------------

Descartes' Rule of Signs

the theorem that can be used to determine the possible numbers of positive real zeros & negative real zeros of a function is called ------------------

critical numbers; test intervals

to solve a polynomial inequality, find the ------------- of the polynomial, & use these numbers to create ---------- for the inequality

imaginary number

a+bi a & b cannot = 0

pure imaginary number

a+bi a=0 b cannot = 0

real number

a+bi b=0

quadratic; parabola

a ---------- function is a second-degree polynomial function, & its graph is called a -----------

standard

a polynomial function is written in ----------- form if its terms are written in descending order of exponents from left to right

nonnegative integer; real

a polynomial function of degree n & leading coefficient a,n is a function of a form f(x)=a,n x^n + a,n-1 x^n-1... (a,n cannot = 0) where n is a ------------- & a1 are -------- numbers

irreducible over the reals

a quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be --------------------.

lower; upper

a real number b is a ---- bound for the real zeros of f if no real zeros are less than b, & is an ------ bound if no real zeros are greater than b

synthetic division

an alternative method to long division of polynomials is called ----------------- in which the divisor must be of the form x-k

dividend; divisor; quotient; remainder; divisor

f(x)/d(x)=q(x)+r(x)/d(x)

dividend; divisor; quotient; remainder

f(x)=d(x)q(x)+r(x)

slant or oblique asymptote

for the rational function given by f(x)=n(x)/d(x) if the degree of n(x) is exactly one more than the degree of d(x), then the graph of f has a -----------

rational functions

functions of the form f(x)=n(x)/d(x), where n(x) & d(x) are polynomials & d(x) is not the zero polynomial are called -----------

principle square

if a is a positive number the -------------- root of the negative number -a is defined as square root of -a = square root of a i

touches; crosses

if a real zero of a polynomial function is of even multiplicity then the graph f ----- the x-axis at x=a, & if it is of odd multiplicity then the graph of f -------- the x-axis at x=a

conjugate

if a+bi is a complex zero of a polynomial with real coefficients, then so is its --------, a-bi

vertical asymptotes

if f(x)---> + or - infinity as x---->a from the left or the right, then x=a is a ------------- of the graph of f

horizontal asymptote

if f(x)--->b as x--> + or - infinity then y=b is a ------- of the graph of f

negative; maximum

if the graph of a quadratic function opens downward, then its leading coefficient is -------- & the vertex of the graph is a --------------

positive; minimum

if the graph of a quadratic function opens upward, then its leading coefficient is -------- & the vertex of the graph is a ---------

factor theorem

the --------- states that a polynomial f(x) has a factor (x-k) if &only if f(k)=0

fundamental theorem of algebra

the ---------- states that if f(x) is a polynomial of degree n (n>0), then f has at least one zero in the complex number system

leading coefficient test

the ----------- is used to determine the left hand & right hand behavior of the graph of a polynomial function

Linear Factorization Theorem

the ----------- states that if f(x) is a polynomial of degree n (n>0) then f has precisely n linear factors f(x)=an(x-c1)(x-c2)....where c1, c2,.....are complex numbers

remainder theorem

the ------------ states that if a polynomial f(x) is divided by x-k the remainder is r=f(k)