Factoring
common factors
A common factor is an expression which can be divided into each term in a polynomial. For example, 3x is a common factor in the polynomial 9x3 - 15x2 + 3x, because 3x divides evenly into each of the three factors. To factor out the common factor we divide each term by that factor, and write the common factor out front; 3x (9x3 /3x - 15x2 /3x + 3x/3x) = 3x(3x2 - 5x + 1).
When a polynomial has a common factor, it is to our advantage to factor it out first. At the very least the remaining polynomial will involve smaller numbers.
Ex. It is possible to factor 15x2 + 45x + 30, without first factoring out the common factor of 15, but if we factor out the 15 first we get, 15(x2 + 3x + 2), a much easier polynomial to factor, 15(x + 2)(x + 1). If we do not factor out the 15 first we get, (15x + 30)(x + 1). This however is not completely factored. In order to have the correct answer we must factor the 15 out of (15x + 30), giving us 15(x +2)(x +1) again.
Sometimes it is impossible to factor the polynomial at all until the common factor is taken out.
Ex. Consider 8x2 - 200. Because this polynomial has only 2 terms, if it can be factored it must be a Difference of Squares, a Difference of Cubes or a Sum of Cubes. Because of the subtraction symbol we know it is not a Sum of Cubes. While 8 is a perfect cube, and x2 is a perfect square, 8x2 is neither. Similiarly, 200 is neither a perfect square or a perfect cube. BUT, if we factor out the common factor of 8, we get 8(x2 - 25). Now x2 is a perfect square and 25 is a perfect square and our answer is 8(x-5)(x+5).
Factor out the common factor, if there is one, in each of the following.
1. 8x4 - 4x3 + 20x2 : solution
2. 25x2 - 100 : solution
3. 5x4 + 135x: solution
1. 8x4 - 4x3 + 20x2
the common factor here is 4x2
4x2 (8x4 /4x2 - 4x3 /4x2 + 20x2 /4x2 )
4x2 (2x2 - x + 5) - the remaining polynomial will not factor.
2. 25x2 - 100
The common factor here is 25
25(25x2 /25 - 100/25)
25(x2 - 4) - the remaining polynomial is a Difference of squares
25(x-2)(x+2)
3. 5x4 + 135x
The common factor here is 5x
5x (5x4 /5x+ 135x /5x )
5x(x3 + 27) - the remaining polynomial here is a Sum of Cubes
5x(x + 3)(x2 - 3x + 9)