Factoring

1. Factoring
1. Factor : When a polynomial is expressed as the product of two or more polynomials, each polynomial is called a factor of the original polynomial.
2. Factoring : The process of expressing a polynomial as the product of its factors. It is essentially the reverse process of expansion.


2. Factoring Using Common Factors
1. Common Factor : A factor that is common to all terms of a polynomial.
2. Factoring Using Common Factors : If a polynomial has a common factor in each term, you can factor it out using the distributive property:
   

Factoring Formulas

1. Perfect Square Formulas
   
   
1. Perfect Square Polynomial : An expression that is a square of a binomial, or a constant multiple of such an expression.
   Examples:
2. Conditions for a Perfect Square Polynomial
1. For to be a perfect square, must satisfy:
2. For to be a perfect square, must satisfy:


2. Difference of Squares
   
   
   
3. Factoring Quadratic Expressions
   
   Steps:
1. Find two numbers whose product equals the constant term.
2. Find two numbers and whose sum equals the coefficient of .
3. Express the quadratic as .


4. Factoring General Quadratic Expressions
   
   Steps:
1. List two numbers whose product equals the coefficient of .
2. List two numbers whose product equals the constant term.
3. Ensure that the diagonal products sum to the coefficient of .
4. Express the quadratic as .
   
   

Factoring Complex Polynomials

1. Factoring Polynomials with Common Parts
   Substitute the common part with a variable and use a factoring formula:
   
   
   
   
2. Factoring Polynomials with Four Terms
   Group terms in pairs to create common factors:
1. Group into terms terms.
2. Group into terms term or term terms to create a difference of squares, .


3. Factoring Polynomials with Five or More Terms
   Organize the terms in descending order of powers of a variable with the lowest degree.

Calculations Using Factoring Formulas

1. Calculating Numbers
   Use factoring formulas to simplify and calculate numbers:
1. Factoring Out Common Factors
   
   Example:
   
2. Using Perfect Square Formula
   
   Example:
   
3. Using Difference of Squares
   
   Example:
   


2. Calculating the Value of an Expression
   Factor the given expression and then substitute the values of the variables to find the result.

Example. Find the value of when .

 Solution