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Algebra

Giac.jl provides powerful symbolic algebra capabilities including polynomial factorization, expansion, simplification, equation solving, and polynomial arithmetic.

Setup

using Giac
using Giac.Commands: factor, expand, simplify, solve, gcd, lcm, quo, rem

@giac_var x y

Polynomial Factorization

Factoring Polynomials

Factor polynomials into irreducible factors using the factor command:

factor(x^2 - 1)
# Output: (x-1)*(x+1)

factor(x^2 - 4)
# Output: (x-2)*(x+2)

factor(x^2 + 2*x + 1)
# Output: (x+1)^2

Factoring Higher-Degree Polynomials

factor(x^3 - 1)
# Output: (x-1)*(x^2+x+1)

factor(x^4 - 1)
# Output: (x-1)*(x+1)*(x^2+1)

Polynomial Expansion

Expanding Products

Expand products and powers of polynomials using the expand command:

expand((x + 1)^2)
# Output: x^2+2*x+1

expand((x + 1)^3)
# Output: x^3+3*x^2+3*x+1

expand((x - 1) * (x + 1))
# Output: x^2-1

Expanding More Complex Expressions

expand((x + 2) * (x - 3) * (x + 1))
# Expands to full polynomial form

expand((x + y)^2)
# Output: x^2+2*x*y+y^2

Simplification

Simplifying Rational Expressions

Simplify expressions using the simplify command:

simplify((x^2-1)/(x-1))
# Output: x+1

simplify((x^3-x)/(x^2-1))
# Output: x

Simplifying Complex Expressions

simplify((x^2+2*x+1)/(x+1))
# Output: x+1

Equation Solving

Solving Polynomial Equations

Solve equations using the solve command:

solve(x^2 - 4, x)
# Output: [-2, 2]

solve(x^2 - 1, x)
# Output: [-1, 1]

solve(x^2 + 2*x + 1, x)
# Output: [-1]

Solving Higher-Degree Equations

solve(x^3 - 1, x)
# Returns all roots including complex

solve(x^2 - 2, x)
# Output includes sqrt(2) and -sqrt(2)

GCD and LCM

Greatest Common Divisor

Find the GCD of polynomials using the gcd command:

gcd(x^2 - 1, x - 1)
# Output: x-1

gcd(x^2 - 4, x^2 - 4*x + 4)
# Output: x-2

Least Common Multiple

Find the LCM of polynomials using the lcm command:

lcm(x - 1, x + 1)
# Output: (x-1)*(x+1) or equivalent

lcm(x^2, x^3)
# Output: x^3

Polynomial Division

Quotient

Compute the quotient of polynomial division using the quo command:

quo(x^3 - 1, x - 1)
# Output: x^2+x+1

quo(x^4, x^2)
# Output: x^2

Remainder

Compute the remainder of polynomial division using the rem command:

rem(x^3, x - 1)
# Output: 1

rem(x^3 + x, x^2 + 1)
# Computes the polynomial remainder

Quotient and Remainder Together

For polynomial division, the relationship dividend = quotient * divisor + remainder always holds:

# For x^3 - 1 divided by x - 1:
# quo(x^3 - 1, x - 1) = x^2 + x + 1
# rem(x^3 - 1, x - 1) = 0
# Verification: (x - 1) * (x^2 + x + 1) = x^3 - 1 ✓

Systems of Equations

Solving Linear Systems

Solve systems of equations by passing lists of equations and variables:

@giac_var x y

# System: x + y = 1, x - y = 0
solve([x + y ~ 1, x - y ~ 0], [x, y])
# Output: [[1/2, 1/2]]

Solving Nonlinear Systems

# System: x^2 + y^2 = 1, x = y
solve([x^2 + y^2 ~ 1, x ~ y], [x, y])
# Returns solutions on the unit circle where x = y

Notes

  • All algebraic operations work symbolically, not numerically
  • The ~ operator creates equations for use with solve
  • For numerical solutions, convert results using to_julia
  • simplify may not always produce the simplest form; try normal for rational simplification
  • factor factors over the rationals by default; use cfactor for complex factorization