Algebraic Expressions Algebraic expressions - preliminaries Evaluating algebraic expressions Simplifying algebraic expressions Expanding algebraic expression by removing parentheses (i.e. brackets)
The square of a binomial - the perfect square trinomial
The square of a trinomial
The cube of binomial
The binomial expansion algorithm Using a variety of methods including combinations of the above to factorize algebraic expressions
Algebraic expressions - preliminaries
An algebraic expression is one or more algebraic terms containing variables and constants connected by mathematical operations. Terms are the elements separated by the plus or minus signs.
In algebraic expressions, variables are letters, such as a,  bc, or x, yz, that can have different values.
Constants are the terms or elements represented only by numbers. Coefficients are the number part of the terms that multiply a variable or powers of a variable.
An algebraic expression consisting of a single term is called a monomial, expression consisting of two terms is binomial, three terms trinomial and an expression with more than three terms is called polynomial
Evaluating algebraic expressions
To evaluate an algebraic expression means to replace (substitute) the variables in the expression with numeric values that are assigned to them and perform the operations in the expression.
 Example: Evaluate the expression:  1 +  a2b + 1/4a4b2    for  a = -1  and  b =  -2. Solution: 1 + (-1)2 ·  (-2) + 1/4 · (-1)4 ·  (-2)2 = 1 - 2 + 1/4 · 1 ·  4 = 0. Or, since  1+ a2b + 1/4a4b2 = (1+ 1/2a2b)2 then, by substituting  a = -1  and  b =  -2,    [1 + 1/2·(-1)2 · (-2)]2 = 0.
Simplifying algebraic expressions
By simplifying an algebraic expression, we mean reducing it in the simplest possible form which mainly involves: multiplication and division, removing (expanding) brackets and collecting (adding and subtracting) like terms.
Like terms are those terms which contain the same powers of same variables and which can only differ in coefficients.
 Examples: a)   - 4a3 + 3a2 + 5a3 - 7a2 = (- 4 + 5) · a3 + (3 - 7) · a2  = a3 - 4a2, b)   (x2  - x + 1) · (x + 1) = x3 - x2 + x + x2  - x + 1 = x3 + 1.
Expanding algebraic expression by removing parentheses (i.e. 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.
The binomial expansion algorithm - the binomial theorem
The binomial expansion of any positive integral power of a binomial, which represents a polynomial with n + 1 terms, or written in the form of the sum formula  is called the binomial theorem.
The binomial coefficients can also be obtained by using Pascal's triangle.
 The triangular array of integers, with 1 at the apex, in which each number is the sum of the two numbers above it in the preceding row, as is shown in the initial segment in the diagram, is called Pascal's triangle. So, for example the last row of the triangle contains the sequence of the coefficients of a binomial of the 5th power.
 n 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 - 1 - - - - - 1
 Example: Using a variety of methods including combinations of the above to factorize algebraic expressions
 Examples: a)  x2 - 2xy + y2 + 2y - 2x = (x - y)2 - 2(x - y) = (x - y)(x - y - 2), b)  x2 - y2 + xz - yz = (x - y)(x + y) + z(x - y) = (x - y)(x + y + z), c)  4x2  - 4xy  + y2  - z2 = (2x - y)2   - z2 = (2x - y - z)(2x - y + z), d)  a3  - 7a + 6 = a3  - a - 6a + 6 = a(a2 -1) - 6(a -1) = (a -1)·[a(a + 1) - 6] = (a -1)(a2 + a - 6) = = (a -1)(a2 + 3a - 2a - 6) = (a -1)[a(a + 3) - 2(a + 3)] = (a -1)(a + 3)(a - 2).   Pre-calculus contents A 