Forms {form, polynomial} are polynomial expressions. Forms can have any variable degree or number {module, polynomial} {modular system}. Forms have finite numbers of basic forms {basis, polynomial}, which make complete systems {Hilbert's basis theorem, form}.
In polynomials, variable has highest exponent {degree, polynomial} {index, polynomial}. For example, polynomial x^2 + 4*x has degree 2.
For one-variable functions, variable power determines graph {graphing}.
Linear function has first power and is straight line, with slope and y-intercept.
Quadratic function has second power and is parabola, with two x-intercepts and one y-intercept.
Cubic equation has third power and has S shape, with three x-intercepts and one y-intercept.
Products of independent and dependent variables, x*y, are hyperbolas.
conics
General functions can add two variables, each raised to second and first power: A * x^2 + B * y^2 + C * x + D * y + E. Functions can equal zero. Equations are ellipses if b^2 - 4*a*c < 0, hyperbolas if b^2 - 4*a*c > 0, and parabolas if b^2 - 4*a*c = 0.
If two polynomials are equal, then coefficients of terms with same variables with same exponents are equal {undetermined coefficients principle} {principle of undetermined coefficients}.
To add polynomials {polynomial addition}, first put terms in simplest form. Then add coefficients of terms that have same variables with same exponents. Sums have same number of terms as total number of different terms in both polynomials.
polynomial division
To divide two polynomials, write polynomials with terms decreasing from highest exponent term. Divide smaller second-polynomial first term into larger first-polynomial first term. Multiply smaller polynomial by quotient. Subtract product from first polynomial. Divide difference by second-polynomial first term, to get new quotient. Then repeat steps. For example, (12*x^2 - x - 6) / (3*x + 2) = (4*x) * (3*x + 2) - 9*x - 6 = (4*x - 3) * (3*x + 2).
polynomial multiplication
To multiply polynomials, multiply each first-polynomial term by each second-polynomial term. Product-term number is number of first-polynomial terms times number of second-polynomial terms. Put terms in simplest form. Add coefficients of terms that have same variables with same exponents.
Products can result in ordered pairs {Cartesian product, function}, denoted [X,Y].
A number or polynomial {factor, polynomial} can divide into another number or polynomial with no remainder. Try to find prime number that factors, try coefficient, try variable, and then try simple polynomial. Different terms can share a prime-factor product {greatest common factor, polynomial}, which divides into terms with no fractional remainder. Linear or quadratic polynomial with real coefficients can have factors with real coefficients.
Polynomials can equal smaller-polynomial products {factoring, polynomial}. For example, a^3 - b^3 factors to (a - b)*(a^2 + a*b + b^2).
difference of squares
Binomials can factor if it they are differences between two squares. For example, x^2 - y^2 factors as (x + y)*(x - y). 9*(x^2) - 64*(y^4) factors as (3*x + 8*(y^2))*(3x - 8*(y^2)).
process
To factor polynomial, first try to find number, coefficient, or variable {monomial factor} that is in all terms. Then try factor with two terms {binomial factor}. First, try binomial whose first term has coefficient that factors highest-power-term coefficient and has highest-power-term variable with no exponent. Second term is number that factors polynomial number term.
process: quadratic trinomial
To factor quadratic trinomials, first place terms in decreasing order of powers. Factor trinomial by highest-power-term coefficient. Try to factor trinomial by variable. Find constant-term numerator and denominator factors. From factors, use two numbers that add to middle-term coefficient. Then factors are (x + number1) and (x + number2).
a*(x^2) + b*x + c factors to a*(x^2 + x*(b/a) + c/a) which factors to (x + c1/a1)*(x + c2/a2), where c = c1*c2, a = a1*a2, b/a = (c1/a1 + c2/a2), and b = c1*a2 + c2*a1.
process: quadrinomial
To factor quadrinomials, first try to find a monomial factor using any term pair. For example, a + b + c + d can factor to e*(f + g) + c + d. Then try to find binomial factor shared by two term pairs. For example, 6*a*x - 2*b - 3*a + 4*b*x factors to 3*a*(2*x - 1) + 2*b*(2*x - 1) which factors to (2*x - 1)*(3*a + 2*b).
process: test binomial
If polynomial has no factors, use test binomial factor. The variable is in highest polynomial term with no exponent. Add constant. For example, x^2 + x + 1 has test factor (x + 1). Divide polynomial by test factor {synthetic division}, to get quotient polynomial and remainder polynomial {remainder theorem}. For example, (x^2 + x + 1)/(x + 1) = x + 1/(x^2 + x + 1).
If remainder is zero, test factor is polynomial factor {factor theorem}. If remainder is zero, negative of constant is a polynomial zero {converse, factor theorem}. For example, (x^2 + 2x + 1)/(x + 1) = x + 1, so remainder is zero, and x is -1.
After dividing power functions or polynomials by modulus, if remainders are the same, the power functions or polynomials are congruent quadratics {quadratic reciprocity law} {law of quadratic reciprocity}, biquadratics {law of biquadratic reciprocity}, and cubics {cubic reciprocity law} {law of cubic reciprocity}.
Functions {homogeneous function} can allow factoring a constant: (k^n) * f(x, y, ...) = f(k*x, k*y, ...), where n is degree, k is constant, f is homogeneous function, and x, y, ... are variables.
Functions {linear function, polynomial} can relate variables with first power.
Variable base can have constant power in function {power function}|. For example, x^3.
Finite complete systems for any-degree binary forms can have rational integral invariants and covariants {Clebsch-Gordan theorem}. Covariants are function projections in binary form.
Forms can have any variable degree or number. Forms can use finite numbers of basic forms, which make complete systems {Hilbert's basis theorem, polynomial} {Hilbert basis theorem, polynomial}.
For all terms, sums of variable exponents can be constant {homogeneous expression}. An example is a^2 * b^1 + a^1 * b^2.
If denominators have no variables, all variable exponents are positive {integral expression}.
Polynomials {perfect power, polynomial} can be nth powers of similar polynomials.
Binomials squared make trinomials {perfect trinomial square}.
Algebraic polynomials {quantic} can have two or more variables, be homogeneous, and be rational integral functions. Quantics can have two variables {binary polynomial}, three variables {ternary polynomial}, four variables {quaternary polynomial}, two degrees {quadric polynomial}, three degrees {cubic polynomial}, four degrees {quartic polynomial}, or five orders or degrees {quintic polynomial}.
Expressions {radical expression} can contain radical signs.
Expressions can have no radical expressions or fractional exponents {rational expression}. Rational expressions can be quotients of two polynomials. Polynomials are rational integral expressions.
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Date Modified: 2022.0225