Algebra¶
This tutorial is a hands-on tour of Mathilda's algebra — the manipulation and simplification of polynomials and rational expressions. You will multiply expressions out and factor them back, pull a polynomial apart to inspect its pieces, reorganise it into Horner or square-free form, run the polynomial division/GCD/resultant toolkit, put fractions over a common denominator and split them up again, and simplify with and without assumptions. Everything here is pure rewriting of exact expressions. Equation solving is a topic in its own right and lives in the next tutorial; the calculus tutorial then builds on exactly this machinery.
Every transcript below was produced by the actual Mathilda binary. Type the
In[...] lines yourself (without the prompt) and you will see the same
Out[...]. Mathilda's output is sometimes ordered differently from a textbook —
it sorts terms into a canonical order, so constants and lower-degree terms tend
to come first — but it is always mathematically correct.
Expand and Factor: two sides of one coin¶
Expand multiplies everything out into a flat sum of terms. Factor runs the
other way, collapsing a polynomial back into a product. They are inverses, and
chaining one into the other is a satisfying way to see both at work.
In[1]:= Expand[(x + 2)(x - 3)]
Out[1]= -6 - x + x^2
In[2]:= Expand[(1 + x)^5]
Out[2]= 1 + 5 x + 10 x^2 + 10 x^3 + 5 x^4 + x^5
In[3]:= Factor[1 + 5 x + 10 x^2 + 10 x^3 + 5 x^4 + x^5]
Out[3]= (1 + x)^5
In[1] multiplies a product of two binomials; In[2] raises a binomial to the
fifth power, producing the row of binomial coefficients 1, 5, 10, 10, 5, 1; and
In[3] factors that same degree-five polynomial straight back into (1 + x)^5.
Expand handles several variables at once, distributing across every term:
In[1]:= Expand[(x + y)^3]
Out[1]= x^3 + 3 x^2 y + 3 x y^2 + y^3
In[2]:= Expand[(a + b + c)^2]
Out[2]= a^2 + 2 a b + b^2 + 2 a c + 2 b c + c^2
Factor is the more impressive of the pair, because finding factors is genuinely
hard. It recognises the standard factorisations and more:
In[1]:= Factor[x^2 - 5 x + 6]
Out[1]= (-3 + x) (-2 + x)
In[2]:= Factor[6 x^2 + 11 x + 3]
Out[2]= (1 + 3 x) (3 + 2 x)
In[3]:= Factor[x^4 - 1]
Out[3]= (-1 + x) (1 + x) (1 + x^2)
In[4]:= Factor[x^6 - 1]
Out[4]= (-1 + x) (1 + x) (1 + x + x^2) (1 - x + x^2)
The quadratic in In[1] splits into two linear factors; In[2] shows Factor
coping with a leading coefficient. The differences of powers in In[3] and
In[4] break into all of their cyclotomic pieces — note that 1 + x^2 is left
intact because it has no real roots.
By default Factor works over the rationals, so an irreducible-over-the-rationals
polynomial is returned unchanged. To factor over a larger number field, name the
algebraic number to adjoin with the Extension option:
In[1]:= Factor[x^2 - 2]
Out[1]= -2 + x^2
In[2]:= Factor[x^2 - 2, Extension -> Sqrt[2]]
Out[2]= (Sqrt[2] + x) (-Sqrt[2] + x)
x^2 - 2 has no rational roots, so In[1] leaves it alone; adjoining Sqrt[2]
in In[2] lets Factor split it into (x - √2)(x + √2).
Factor has a few specialised cousins. FactorSquareFree only separates out
repeated factors, leaving the square-free part as one block instead of
factoring it fully — cheaper than Factor, and exactly what you want before
square-free–based algorithms like partial fractions or symbolic integration:
In[1]:= FactorSquareFree[(x - 1)^2 (x + 1)^3]
Out[1]= (-1 + x)^2 (1 + x)^3
In[2]:= FactorSquareFree[x^4 + 2 x^3 + 2 x^2 + 2 x + 1]
Out[2]= (1 + x)^2 (1 + x^2)
In[2] finds the repeated linear factor (1 + x)^2 but leaves the remaining
square-free 1 + x^2 un-factored. FactorTerms goes the other way and pulls out
only the overall numeric content, leaving the polynomial part alone, while
FactorTermsList returns that content split into integer and content parts
alongside the remaining polynomial:
In[1]:= FactorTerms[2 x^2 + 4 x + 6]
Out[1]= 2 (3 + 2 x + x^2)
In[2]:= FactorTermsList[2 x^2 + 4 x + 6, x]
Out[2]= {2, 1, 3 + 2 x + x^2}
In[1] factors the common 2 out front; In[2] reports the same content as the
first list element and the monic remainder as the last.
Looking inside a polynomial¶
A polynomial is just an expression, so you can interrogate its structure. The most direct questions are "what is the coefficient of this power?" and "what are all the coefficients?":
In[1]:= Coefficient[x^2 + 3 x + 2, x]
Out[1]= 3
In[2]:= Coefficient[3 x^2 + 5 x + 7, x, 2]
Out[2]= 3
In[3]:= CoefficientList[(x + 1)^3, x]
Out[3]= {1, 3, 3, 1}
Coefficient[poly, x] returns the coefficient of the first power of x (In[1]),
while the three-argument form Coefficient[poly, x, n] picks out the coefficient
of x^n (In[2] reads off the leading 3). CoefficientList returns all the
coefficients at once, lowest power first, so In[3] gives the cube's row of
coefficients as a list. These compose with Expand — to read the coefficient of
x^3 in (1 + x)^5, expand first and then ask:
To discover which symbols even appear, use Variables:
The dual operation to expanding is grouping. Collect[expr, x] gathers all the
terms that share a power of x, folding their coefficients together:
In[1]:= Collect[x y + x z + a x, x]
Out[1]= x (a + y + z)
In[2]:= Collect[a x^2 + b x^2 + x, x]
Out[2]= x + (a + b) x^2
In[1] factors the common x out of three terms; In[2] keeps the powers of
x separate but merges the two x^2 coefficients into a + b. Collect is the
tool of choice when you want a polynomial organised by one variable while
treating the others as parameters.
PolynomialQ is the predicate that decides whether an expression is a
polynomial in the given variable — the gatekeeper before you reach for any of the
tools above:
1/x is x^(-1), a negative power, so In[2] is not a polynomial in x. Two
final reshaping tools change a polynomial's form without changing its value.
HornerForm rewrites it in nested Horner form, the arrangement that evaluates in
the fewest multiplications, and Decompose finds a functional decomposition —
expressing the polynomial as p(q(x)) for smaller p and q:
In[1]:= HornerForm[1 + 2 x + 3 x^2 + 4 x^3]
Out[1]= 1 + x (2 + x (3 + 4 x))
In[2]:= Decompose[x^4 + 2 x^2 + 3, x]
Out[2]= {3 + 2 x + x^2, x^2}
In[1] nests the four coefficients so the cubic costs only three multiplies to
evaluate. In[2] discovers that x^4 + 2 x^2 + 3 is (u^2 + 2 u + 3) composed
with u = x^2; the returned list {p, q} reads outermost-first, so the original
polynomial is p applied to q.
The polynomial toolkit¶
Polynomials support their own arithmetic. PolynomialGCD finds the greatest
common divisor of two polynomials, exactly as GCD does for integers, and
PolynomialQuotient/PolynomialRemainder perform division with remainder:
In[1]:= PolynomialGCD[x^2 - 1, x^2 - 3 x + 2]
Out[1]= -1 + x
In[2]:= PolynomialQuotient[x^2 - 1, x - 1, x]
Out[2]= 1 + x
In[3]:= PolynomialRemainder[x^2 + 1, x - 1, x]
Out[3]= 2
Both x^2 - 1 and x^2 - 3x + 2 share the factor x - 1, which In[1] finds.
In[2] divides x^2 - 1 by x - 1 exactly (quotient x + 1, no remainder),
while In[3] divides x^2 + 1 by x - 1 and reports the leftover 2 —
precisely the value of x^2 + 1 at x = 1, as the remainder theorem promises.
Two further tools answer questions about roots without computing them.
Resultant[p, q, x] is zero exactly when p and q share a common root, and
Discriminant[p, x] is zero exactly when p has a repeated root:
In[1]:= Resultant[x^2 - 1, x - 1, x]
Out[1]= 0
In[2]:= Discriminant[x^2 + b x + c, x]
Out[2]= b^2 - 4 c
In[1] is zero because x = 1 is a root of both polynomials. In[2] recovers
the familiar quadratic discriminant b² − 4c symbolically — the very quantity
whose sign tells you how many real roots x² + bx + c has.
The GCD comes with the rest of the divisibility family. PolynomialLCM is the
least common multiple, and PolynomialExtendedGCD returns not just the GCD but
the Bézout cofactors {a, b} with a p + b q = gcd:
In[1]:= PolynomialLCM[x^2 - 1, x^2 - 3 x + 2]
Out[1]= (1 + x) (2 - 3 x + x^2)
In[2]:= PolynomialExtendedGCD[x^2 - 1, x^2 - 3 x + 2, x]
Out[2]= {-1 + x, {1/3, -1/3}}
In[1] multiplies the two quadratics' shared factor in only once. In In[2] the
first element -1 + x is the GCD and {1/3, -1/3} are the cofactors: one third
of the first polynomial minus one third of the second is exactly x - 1.
PolynomialMod reduces every coefficient modulo an integer — the entry point to
arithmetic over finite fields:
5 ≡ 2 and 7 ≡ 1 modulo 3, so the coefficients collapse accordingly.
IrreduciblePolynomialQ asks whether a polynomial factors at all over the
rationals — the predicate underlying Factor:
In[1]:= IrreduciblePolynomialQ[x^2 + 1]
Out[1]= True
In[2]:= IrreduciblePolynomialQ[x^2 - 1]
Out[2]= False
x^2 + 1 has no rational roots and stays whole; x^2 - 1 splits into
(x - 1)(x + 1). Resultant is the top of a whole ladder of subresultants.
Subresultants returns their leading coefficients (the principal subresultant
coefficients) and SubresultantPolynomials returns the polynomials themselves —
the same remainder sequence a Euclidean GCD walks, but computed without fractions:
In[1]:= Subresultants[x^3 + x + 1, x^2 + x + 1, x]
Out[1]= {3, 1, 1}
In[2]:= SubresultantPolynomials[x^3 + x + 1, x^2 + x + 1, x]
Out[2]= {3, 2 + x, 1 + x + x^2}
The first entry of In[1], 3, is the resultant itself; being non-zero, it
confirms the two cubics share no common root. In[2] lists the corresponding
subresultant polynomials, ending in the higher of the two inputs.
For several polynomials in several variables at once, GroebnerBasis
computes a Gröbner basis — a canonical generating set for the ideal the
polynomials define. It is the multivariate generalisation of both PolynomialGCD
(one variable) and Gaussian elimination (linear systems), and it is the engine
behind solving polynomial systems:
The basis In[1] is triangular: the first polynomial 2 y^2 - 1 involves only
y, so it can be solved on its own, and then x - y pins down x. We will lean
on this structure heavily in the applications below.
Rational expressions¶
A ratio of polynomials is a rational expression, and Mathilda gives you precise
control over its shape without ever changing its value. Together puts a sum of
fractions over a single common denominator:
In[1]:= Together[1/x + 1/(x + 1)]
Out[1]= (1 + 2 x)/(x + x^2)
In[2]:= Together[a/b + c/d]
Out[2]= (b c + a d)/(b d)
Apart does the reverse, splitting one fraction into a sum of simpler partial
fractions — the decomposition you learned for integrating rational functions by
hand:
In[1]:= Apart[(2 x + 3)/((x + 1)(x + 2))]
Out[1]= 1/(1 + x) + 1/(2 + x)
In[2]:= Apart[(x^3 + x + 1)/(x^2 - 1)]
Out[2]= x + 1/2/(1 + x) + 3/2/(-1 + x)
In[1] breaks a proper fraction into two simple ones. In[2] first divides out
a polynomial part (x, because the numerator's degree exceeds the
denominator's) and then expresses the remaining proper fraction in partial
fractions.
Cancel removes common factors between numerator and denominator, and
Numerator/Denominator extract the two halves:
In[1]:= Cancel[(x^2 - 1)/(x - 1)]
Out[1]= 1 + x
In[2]:= Numerator[(x + 1)/(x - 1)]
Out[2]= 1 + x
In[3]:= Denominator[(x + 1)/(x - 1)]
Out[3]= -1 + x
In In[1], the numerator factors as (x - 1)(x + 1) and the x - 1 cancels
against the denominator, leaving 1 + x. When you want to expand one half of a
fraction while leaving the other in factored form, reach for ExpandNumerator
and ExpandDenominator:
In[1]:= ExpandNumerator[(x + 1)^2/(x - 1)^2]
Out[1]= (1 + 2 x + x^2)/(-1 + x)^2
In[2]:= ExpandDenominator[(x + 1)^2/(x - 1)^2]
Out[2]= (1 + x)^2/(1 - 2 x + x^2)
In[1] multiplies out the numerator only; In[2] does the same for the
denominator only — finer control than Expand, which would flatten both at once.
A different normalisation applies to powers and logarithms. PowerExpand rewrites
(a b)^n as a^n b^n and Log[a b] as Log[a] + Log[b], treating every base
as positive — a deliberately un-conditional expansion that is only valid under
positivity assumptions, so use it knowingly:
In[1]:= PowerExpand[Sqrt[x^2 y]]
Out[1]= x Sqrt[y]
In[2]:= PowerExpand[Log[a b]]
Out[2]= Log[a] + Log[b]
In[1] pulls x out of the radical (which assumes x ≥ 0), and In[2] splits
the logarithm of a product into a sum of logarithms.
Simplify and assumptions¶
Simplify is the all-purpose tool: it tries a battery of transformations —
factoring, cancelling, expanding, applying identities — and keeps whichever
result is smallest. It subsumes much of what the previous sections did manually:
In[1]:= Simplify[(x^2 - 1)/(x - 1)]
Out[1]= 1 + x
In[2]:= Simplify[(x^3 - 1)/(x - 1)]
Out[2]= 1 + x + x^2
In[3]:= Simplify[Sin[x]^2 + Cos[x]^2]
Out[3]= 1
In[1] and In[2] cancel removable factors, and In[3] applies the
Pythagorean trigonometric identity — a simplification no amount of factoring
alone would find.
Some simplifications are only valid under extra conditions. For instance
Sqrt[x^2] equals x only when x is non-negative (otherwise it is -x).
Supply such facts as a second argument to Simplify, or wrap a whole block in
Assuming:
Both say the same thing: given that x > 0, Sqrt[x^2] simplifies to x.
Without the assumption Mathilda leaves the expression alone, because the
simplification would be wrong for negative x.
Putting the toolkit to work¶
The real power of the algebra toolkit shows up when several tools combine to crack a problem that has no obvious direct command. Here are three.
Implicitising a parametric curve with Resultant¶
A curve is often given parametrically, as x = f(t), y = g(t). To find the
single polynomial equation F(x, y) = 0 that the points (x, y) satisfy — its
implicit equation — you must eliminate the parameter t. That is exactly what
Resultant does: the resultant of x - f(t) and y - g(t) with respect to t
vanishes precisely when both share a common t, i.e. for points actually on the
curve. Take the parametrisation x = t^2 - 1, y = t^3 - t:
So every point of the curve satisfies y^2 = x^3 + x^2 — the nodal cubic, a
classic curve that crosses itself at the origin. We started from two cubics in a
hidden parameter and recovered one clean equation in x and y alone, with no
trigonometry or substitution by hand.
Solving a symmetric polynomial system with GroebnerBasis¶
Suppose three unknowns satisfy the power-sum conditions
This is a genuinely nonlinear system. Computing a Gröbner basis with the
variables ordered so that z is eliminated last triangularises it:
In[1]:= GroebnerBasis[{x + y + z - 6, x^2 + y^2 + z^2 - 14, x^3 + y^3 + z^3 - 36}, {x, y, z}]
Out[1]= {-6 + 11 z - 6 z^2 + z^3, 11 - 6 y + y^2 - 6 z + y z + z^2, -6 + x + y + z}
The first basis element involves only z: it is z^3 - 6 z^2 + 11 z - 6. Factor
it and the roots fall out:
So z ∈ {1, 2, 3}, and by the symmetry of the original equations {x, y, z} is
just a permutation of {1, 2, 3}. The Gröbner basis turned an intimidating
nonlinear system into a single one-variable cubic — the same elimination strategy
Gaussian elimination uses for linear systems, generalised to polynomials.
Building a minimal polynomial for a nested radical¶
How do you find a polynomial equation with integer coefficients satisfied by a
messy nested radical such as √(2 + √3)? MinimalPolynomial answers directly:
So √(2 + √3) is a root of x⁴ - 4 x² + 1. We can confirm it by substituting the
radical back in and simplifying to zero:
The expression collapses to 0, proving the radical really does satisfy that
quartic — a degree-4 algebraic number captured exactly, with no decimals.
Where to next¶
You can now expand and factor polynomials, dissect them with Coefficient and
Collect, reshape them into Horner and square-free form, run the full
division/GCD/resultant/Gröbner toolkit, reshape rational expressions with
Together/Apart/Cancel, and simplify with and without assumptions. This is
the algebraic foundation the rest of Mathilda's symbolic mathematics rests on.
- 7. Solutions of equations — turn this
machinery on equations and systems:
Solve,Reduce,Roots, and friends build directly on the factoring, resultant, and Gröbner tools above. - 8. Calculus — put this algebra to work: differentiate,
integrate (leaning on
Apartfor rational functions), expand functions as power series, take limits, evaluate sums, and solve problems numerically. - Function reference — the full catalogue of
built-in functions. The
arithmetic,
algebra, and
simplification sections cover every
function used above, including the complete options for
FactorandSimplify.