Algebraic Expressions

Expanding algebraic expression by removing parentheses (i.e. brackets)
The square of a binomial (or binomial square )
Cube of a binomial
The square of a trinomial
Expanding algebraic expression by removing parentheses (brackets)
The operation of multiplying out algebraic expressions that involve parentheses using the distributive property is often described as expanding the brackets.
Some important binomial products like perfect squares, and difference of two squares are used to help with factoring algebraic expressions.
 Examples: a)   (a - b)2 = (a - b) · (a - b) = a2 - ab - ab + b2 = a2 - 2ab + b2, b)   (a - b) · (a + b) = a2 - ab + ab - b2 = a2 - b2, c)   (x + y) · (x2 - xy + y2) = x3 - x2y + xy2 + x2y  - xy2 + y3 = x3 + y3.
The square of a binomial (or binomial square)
To the square of the first term add twice the product of the two terms and the square of the last term.
 Examples: a)   (a + b)2 = (a + b) · (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2, b)   (2x + 3)2 = (2x)2 + 2 · (2x) · 3 + 32 = 4x2 + 12x + 9, c)   (x - 2y)2 = x2  + 2 · x · (-2y) + (-2y)2 = x2 - 4xy + 4y2.
Squaring trinomial (or trinomial square)
To the sum of squares of the 1st, the 2nd and the 3rd term add, twice the product of the 1st and the 2nd term, twice the product of the 1st and the 3rd term, and twice the product of the 2nd and the 3rd term.
 Examples: a)  (x2 - 2x + 5)2 = (x2)2 + (2x)2 + 52 + 2 · x2 · (-2x) + 2 · x2 · 5 + 2 · (-2x) · 5 = = x4 + 4x2 + 25 - 4x3 + 10x2 - 20x = x4  - 4x3 + 14x2 - 20x + 25, b)  (a3 - a2b - 3ab2)2 = (a3)2 + (a2b)2 + (3ab2)2 + 2a3 (-a2b) + 2a3 (-3ab2) + 2(-a2b) (-3ab2) = = a6 + a4b2 + 9a2b4 - 2a5b - 6a4b2 + 6a3b3  = a6 - 5a4b2 + 9a2b4 - 2a5b + 6a3b3.
Cube of a binomial
To the cube of the first term add, three times the product of the square of the first term and the last term, three times the product of the first term and the square of the last term, and the cube of the last term.
 Examples: a)  (a - b)3 = (a - b)2 · (a - b) = (a2 - 2ab + b2) · (a - b) = = a3 - 2a2b + ab2 - a2b + 2ab2 - b3 = a3 - 3a2b + 3ab2 - b3, b)  (x - 2)3 = x3 + 3 · x2 · (-2) + 3 · x · (-2)2  + (-2)3 = x3  - 6x2 + 12x - 8, c)  (2x + y)3 = (2x)3 + 3 · (2x)2 · y + 3 · (2x) · y2  + y3 = 8x3 + 12x2y + 6xy2  + y3.
Beginning Algebra Contents B