a(b + c) = ab + ac
Distributive property: Multiply a by each term inside the parentheses and then add.
a(b – c) = ab – ac
Distributive property with subtraction: Multiply a by each term inside the parentheses, subtract.
(x + y)² = x² + y² + 2xy
Square of a sum: Square each term and add twice the product of the two terms.
(x – y)² = x² + y² – 2xy
Square of a difference: Square each term and subtract twice the product of the two terms.
(x – y)(x + y) = x² – y²
Difference of squares: Multiply conjugate binomials to get the difference of their squares.
(x + y)³ = x³ + 3x²y + 3xy² + y³
Cube of a sum: Use binomial expansion to cube a sum.
(x – y)³ = x³ – 3x²y + 3xy² – y³
Cube of a difference: Use binomial expansion for the difference.
(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
Fourth power of a sum: Binomial expansion.
(x – y)⁴ = x⁴ – 4x³y + 6x²y² – 4xy³ + y⁴
Fourth power of a difference: Binomial expansion.
(x + y)² – (x – y)² = 4xy &&
Difference of squares of sums and differences: This simplifies to 4 times the product of x and y.
(x + y)² + (x – y)² = 2(x² + y²) or (x² + y²)=1/2[(x + y)² + (x – y)²]
x² + y² = (x + y)² – 2xy
Expressing sum of squares: Rewrite using binomial expansion.
(x + y)² = x² + 2xy + y²
Square of a sum: Expand binomial.
(x – y)² = x² – 2xy + y²
Square of a difference: Expand binomial.
x⁴ – y⁴ = (x² + y²)(x² – y²)
Difference of fourth powers: Factor into the product of sum and difference of squares.
x⁴ – y⁴ = (x² + y²)(x + y)(x – y)
Alternative factorization of difference of fourth powers.
(x + y)³ + (x – y)³ = 2x³ + 6xy²
Sum of cubes of sum and difference.
(x + y)³ – (x – y)³ = 6x²y + 2xy²
Difference of cubes of sum and difference.
x³ + y³ = (x + y)(x² – xy + y²)
Sum of cubes: Factor into a product.
x³ – y³ = (x – y)(x² + xy + y²)
Difference of cubes: Factor into a product.
(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
Fourth power expansion.
(x + y)²(x – y)² = (x² – y²)²
Product of squares: Simplifies to the square of the difference of squares.
x⁴ + y⁴ = (x² + y²)² – 2x²y²
Sum of fourth powers: Expressed in terms of squares.
(x + y)⁴ + (x – y)⁴ = 2x⁴ + 12x²y² + 2y⁴
Sum of fourth powers of sum and difference.
x⁴ – y⁴ = (x² + y²)(x + y)(x – y)
Difference of fourth powers.
x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
Sum of cubes minus 3 times the product.
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
Square of a trinomial.
(x + y + z)³ = x³ + y³ + z³ + 3x²y + 3x²z + 3y²x + 3y²z + 3z²x + 3z²y + 6xyz
Cube of a sum of three variables.
[x + y + z]² = x² + y² + z² + 2xy + 2yz + 2zx
if x+y+z=0 then the right side value becomes 0
(x + y + z)³ = x³ + y³ + z³ + 3x²y + 3x²z + 3y²x + 3y²z + 3z²x + 3z²y + 6xyz
Cube of trinomial.
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
Repeated binomial square expansion.
x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx) if x + y + z=0 then x³ + y³ + z³=3xyz
Some Important Rules for Simplifying Equations or Quick Answer