A polynomial is a sum of terms which are coefficients times various powers of a “base” variable. For example, ‘2 x^2 + 3 x - 4’ is a polynomial in ‘x’. Some formulas can be considered polynomials in several different variables: ‘1 + 2 x + 3 y + 4 x y^2’ is a polynomial in both ‘x’ and ‘y’. Polynomial coefficients are often numbers, but they may in general be any formulas not involving the base variable.
The a f (
factor] command factors a
polynomial into a product of terms. For example, the polynomial
‘x^3 + 2 x^2 + x’ is factored into ‘x*(x+1)^2’. As another
example, ‘a c + b d + b c + a d’ is factored into the product
‘(a + b) (c + d)’.
Calc currently has three algorithms for factoring. Formulas which are linear in several variables, such as the second example above, are merged according to the distributive law. Formulas which are polynomials in a single variable, with constant integer or fractional coefficients, are factored into irreducible linear and/or quadratic terms. The first example above factors into three linear terms (‘x’, ‘x+1’, and ‘x+1’ again). Finally, formulas which do not fit the above criteria are handled by the algebraic rewrite mechanism.
Calc’s polynomial factorization algorithm works by using the general root-finding command (a P) to solve for the roots of the polynomial. It then looks for roots which are rational numbers or complex-conjugate pairs, and converts these into linear and quadratic terms, respectively. Because it uses floating-point arithmetic, it may be unable to find terms that involve large integers (whose number of digits approaches the current precision). Also, irreducible factors of degree higher than quadratic are not found, and polynomials in more than one variable are not treated. (A more robust factorization algorithm may be included in a future version of Calc.)
The rewrite-based factorization method uses rules stored in the variable
FactorRules. See Rewrite Rules, for a discussion of the
operation of rewrite rules. The default
FactorRules are able
to factor quadratic forms symbolically into two linear terms,
‘(a x + b) (c x + d)’. You can edit these rules to include other
cases if you wish. To use the rules, Calc builds the formula
‘thecoefs(x, [a, b, c, ...])’ where
x is the polynomial
base variable and
b, etc., are polynomial coefficients
(which may be numbers or formulas). The constant term is written first,
i.e., in the
a position. When the rules complete, they should have
changed the formula into the form ‘thefactors(x, [f1, f2, f3, ...])’
fi should be a factored term, e.g., ‘x - ai’.
Calc then multiplies these terms together to get the complete
factored form of the polynomial. If the rules do not change the
thecoefs call to a
thefactors call, a f leaves the
polynomial alone on the assumption that it is unfactorable. (Note that
the function names
thefactors are used only
as placeholders; there are no actual Calc functions by those names.)
The H a f [
factors] command also factors a polynomial,
but it returns a list of factors instead of an expression which is the
product of the factors. Each factor is represented by a sub-vector
of the factor, and the power with which it appears. For example,
‘x^5 + x^4 - 33 x^3 + 63 x^2’ factors to ‘(x + 7) x^2 (x - 3)^2’
in a f, or to ‘[ [x, 2], [x+7, 1], [x-3, 2] ]’ in H a f.
If there is an overall numeric factor, it always comes first in the list.
factors allow a second argument
when written in algebraic form; ‘factor(x,v)’ factors ‘x’ with
respect to the specific variable ‘v’. The default is to factor with
respect to all the variables that appear in ‘x’.
The a c (
collect] command rearranges a
formula as a
polynomial in a given variable, ordered in decreasing powers of that
variable. For example, given ‘1 + 2 x + 3 y + 4 x y^2’ on
the stack, a c x would produce ‘(2 + 4 y^2) x + (1 + 3 y)’,
and a c y would produce ‘(4 x) y^2 + 3 y + (1 + 2 x)’.
The polynomial will be expanded out using the distributive law as
necessary: Collecting ‘x’ in ‘(x - 1)^3’ produces
‘x^3 - 3 x^2 + 3 x - 1’. Terms not involving ‘x’ will
not be expanded.
The “variable” you specify at the prompt can actually be any expression: a c ln(x+1) will collect together all terms multiplied by ‘ln(x+1)’ or integer powers thereof. If ‘x’ also appears in the formula in a context other than ‘ln(x+1)’, a c will treat those occurrences as unrelated to ‘ln(x+1)’, i.e., as constants.
The a x (
expand] command expands an
expression by applying the distributive law everywhere. It applies to
products, quotients, and powers involving sums. By default, it fully
distributes all parts of the expression. With a numeric prefix argument,
the distributive law is applied only the specified number of times, then
the partially expanded expression is left on the stack.
The a x and j D commands are somewhat redundant. Use a x if you want to expand all products of sums in your formula. Use j D if you want to expand a particular specified term of the formula. There is an exactly analogous correspondence between a f and j M. (The j D and j M commands also know many other kinds of expansions, such as ‘exp(a + b) = exp(a) exp(b)’, which a x and a f do not do.)
Calc’s automatic simplifications will sometimes reverse a partial expansion. For example, the first step in expanding ‘(x+1)^3’ is to write ‘(x+1) (x+1)^2’. If a x stops there and tries to put this formula onto the stack, though, Calc will automatically simplify it back to ‘(x+1)^3’ form. The solution is to turn simplification off first (see Simplification Modes), or to run a x without a numeric prefix argument so that it expands all the way in one step.
The a a (
apart] command expands a
rational function by partial fractions. A rational function is the
quotient of two polynomials;
apart pulls this apart into a
sum of rational functions with simple denominators. In algebraic
apart function allows a second argument that
specifies which variable to use as the “base”; by default, Calc
chooses the base variable automatically.
The a n (
attempts to arrange a formula into a quotient of two polynomials.
For example, given ‘1 + (a + b/c) / d’, the result would be
‘(b + a c + c d) / c d’. The quotient is reduced, so that
a n will simplify ‘(x^2 + 2x + 1) / (x^2 - 1)’ by dividing
out the common factor ‘x + 1’, yielding ‘(x + 1) / (x - 1)’.
The a \ (
pdiv] command divides
two polynomials ‘u’ and ‘v’, yielding a new polynomial
‘q’. If several variables occur in the inputs, the inputs are
considered multivariate polynomials. (Calc divides by the variable
with the largest power in ‘u’ first, or, in the case of equal
powers, chooses the variables in alphabetical order.) For example,
dividing ‘x^2 + 3 x + 2’ by ‘x + 2’ yields ‘x + 1’.
The remainder from the division, if any, is reported at the bottom
of the screen and is also placed in the Trail along with the quotient.
pdiv in algebraic notation, you can specify the particular
variable to be used as the base:
pdiv is given only two arguments (as is always the case with
the a \ command), then it does a multivariate division as outlined
The a % (
prem] command divides
two polynomials and keeps the remainder ‘r’. The quotient
‘q’ is discarded. For any formulas ‘a’ and ‘b’, the
results of a \ and a % satisfy ‘a = q b + r’.
(This is analogous to plain \ and %, which compute the
integer quotient and remainder from dividing two numbers.)
The a / (
divides two polynomials and reports both the quotient and the
remainder as a vector ‘[q, r]’. The H a / [
command divides two polynomials and constructs the formula
‘q + r/b’ on the stack. (Naturally if the remainder is zero,
this will immediately simplify to ‘q’.)
The a g (
pgcd] command computes
the greatest common divisor of two polynomials. (The GCD actually
is unique only to within a constant multiplier; Calc attempts to
choose a GCD which will be unsurprising.) For example, the a n
command uses a g to take the GCD of the numerator and denominator
of a quotient, then divides each by the result using a \. (The
definition of GCD ensures that this division can take place without
leaving a remainder.)
While the polynomials used in operations like a / and a g often have integer coefficients, this is not required. Calc can also deal with polynomials over the rationals or floating-point reals. Polynomials with modulo-form coefficients are also useful in many applications; if you enter ‘(x^2 + 3 x - 1) mod 5’, Calc automatically transforms this into a polynomial over the field of integers mod 5: ‘(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)’.
Congratulations and thanks go to Ove Ewerlid
ewerlid@mizar.DoCS.UU.SE), who contributed many of the
polynomial routines used in the above commands.
See Decomposing Polynomials, for several useful functions for extracting the individual coefficients of a polynomial.