new math

complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

cubic function

A cubic function is one in the form f(x) = ax3 + bx2 + cx + d. The "basic" cubic function, f(x) = x3, is graphed below. The function of the coefficient a in the general equation is to make the graph "wider" or "skinnier", or to reflect it (if negative): The constant d in the equation is the y-intercept of the graph.

quartic function

A fourth degree polynomial is called a quartic and is a function, f, with rule.

coefficient

A number or symbol multiplied with a variable or an unknown quantity in an algebraic term. For example, 4 is the coefficient in the term 4x, and x is the coefficient in x(a + b). A numerical measure of a physical or chemical property that is constant for a system under specified conditions.

Polynomial function

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a polynomial, and define its degree. 2. What is a polynomial? A polynomial of degree n is a function of the form.

quadratic function

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

factors

Factors. A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. All numbers have a factor of one since one multiplied by any number equals that number.

end behavior

Graph C: up on the left, down on the right Graph D: up on both ends The important things to consider are the sign and the degree of the leading term. The exponent says that this is a degree-4 polynomial, so the graph will behave roughly like a quadratic: up on both ends or down on both ends. Since the sign on the leading coefficient is negative, the graph will be down on both ends. (The actual value of the negative coefficient, -3 in this case, is actually irrelevant for this problem. All I need is the "minus" part of the leading coefficient.) Clearly Graphs A and C represent odd-degree polynomials, since their two ends head off in opposite directions. Graph D shows both ends passing through the top of the graphing box, just like a positive quadratic would. The only graph with both ends down is: Graph B Describe the end behavior of f(x) = 3x7 + 5x + 1004 This polynomial is much too large for me to view in the standard screen on my graphing calculator, so either I can waste a lot of time fiddling with WINDOW options, or I can quickly use my knowledge of end behavior. This function is an odd-degree polynomial, so the ends go off in opposite directions, just like every cubic I've ever graphed. A positive cubic enters the graph at the bottom, down on the left, and exits the graph at the top, up on the right. Since the leading coefficient of this odd-degree polynomial is positive, then its end-behavior is going to mimic a positive cubic.

synthetic division

In algebra, synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the form x - a,\ but the method generalizes to division by any monic polynomial, and to any polynomial. The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, preventing sign errors. Synthetic division for linear denominators is also called division through Ruffini's rule.

vertex

In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.

quotient

In mathematics, a quotient (from Latin: quotiens "how many times", pronounced ˈkwoʊʃənt) is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor.

complex conjugate

In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. For example, the complex conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of is .

complex conjugate zeros

In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P.[1] It follows from this (and the fundamental theorem of algebra), that if the degree of a real polynomial is odd, it must have at least one real root.[2] That fact can also be proven by using the intermediate value theorem.

complx plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis.

imaginary unit

The "unit" Imaginary Number (the equivalent of 1 for Real Numbers) is √(-1) (the square root of minus one). In mathematics we use i (for imaginary) but in electronics they use j (because "i" already means current, and the next letter after i is j).

degree of a polynomial

The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its canonical form consisting of a linear combination of monomials. The degree of a term is the sum of the exponents of the variables that appear in it.

Parabola

The graph of a quadratic function

repeated zeros

The zeros arising from repeated factors of a polynomial function are called repeated zeros. Example: f(x) = x2 - 6x + 9. Set f(x) to zero, factor and solve.

remainder

a part, number, or quantity that is left over.

continuous function

can be formally defined as a function where the pre-image of every open set in is open in . More concretely, a function in a single variable is said to be continuous at point if. 1. is defined, so that is in the domain of . 2. exists for in the domain of .

local extremum

finition Let S be the domain of f such that c is an element of S. Then, 1) f(c) is a local maximum value of f if there exists an interval (a,b) containing c such that f(c) is the maximum value of f on (a,b)∩S. 2) f(c) is a local minimum value of f if there exists an interval (a,b) containing c such that f(c) is the minimum value of f on (a,b)∩S. 3) f(c) is a local extreme value of f if it is either a local maximum or local minimum value. 18B Local Extrema 3 How do we find the local extrema? First Derivative Test Let f be continuous on an open interval (a,b)

leading coefficient

the coefficient of the term of highest degree in a given polynomial. 5 is the leading coefficient in 5 x 3+ 3 x 2− 2 x + 1.

real part of a complex numberr

where z^_ is the complex conjugate of z. The real part is implemented in the Wolfram Language as Re[z]. A nonzero complex number with zero real part is called an imaginary number or sometimes, for emphasis, a purely imaginary number.