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}.
3-Algebra-Function-Kinds-Polynomial
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225