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.
Mathematical Sciences>Algebra>Function>Kinds>Polynomial>Operations
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Date Modified: 2022.0224