-- Factoring polynomial expressions is not quite the sameas factoring numbers, but the concept is very similar.When factoring numbers or factoring polynomials, you arefinding numbers or polynomials that divide out evenly fromthe original numbers or polynomials. But in the case ofpolynomials, you are dividing numbers and variables outof expressions, not just dividing numbers out of numbers.
1: One common factor. a x + a y = a (x + y)2: Sevearl grouped common factor. a x + a y + b x + b y = a(x + y) + b(x + y) = (a +b ) (x + y)3: Difference of two squares (1). x 2 - y 2 = (x + y)(x - y)4: Difference of two squares (2). (x + y) 2 - z 2 = (x + y + z)(x + y - z)5: Sum of two cubes. x 3 + y 3 = (x + y)(x 2 - x y + y 2)6: Difference of two cubes. x 3 - y 3 = (x - y)(x 2 + x y + y 2)7: Difference of fourth powers. x 4 - y 4 = (x 2 - y 2)(x 2 + y 2) = (x + y)(x - y)(x 2 + y 2)8: Perfet square x 2 + 2xy + y 2 = (x + y) 29: Perfet square x 2 - 2xy + y 2 = (x - y) 210: Perfect cube x 3 + 3x 2y + 3xy 2 + y 3 = (x + y) 311: Perfect cube x 3 - 3x 2y + 3xy 2 - y 3 = (x - y) 3
Previously, you have simplified expressions by distributing through parentheses, such as:Example : 2(x + 3) = 2(x) + 2(3) = 2x + 6The trick is to see what can be factored out of every term in the expression. Warning: Dont make the mistake of thinking that "factoring" means "dividing something off and making it magically disappear". Remember that "factoring" means "dividing out and putting in front of the parentheses". Nothing "disappears" when you factor; things merely get rearranged.
Any number or variable that is a factor of both terms in a binomial can be factored out. For example, the binomial 3x^3 + 6x can be factored to (3x)(x^2 + 2) because 3x is a factor of 3x^2 (which is 3x * x^2) and of 6 (which is 3x * 2).
Find the greatest common factor (GCF) of both terms. The greatest common factor is the largest value that can be factored out of both terms. In the expression 6y^2 - 24, 3 is a common factor of both terms, but it is not the GCF. Six is the GCF, since both numbers can be divided by 6. Factoring it out, we get 6(y^2 - 4).
Find out if you have a difference of squares. A difference of squares is a variable squared minus a constant, like y^2 - 4. If you have factored out the GCF and dont have a minus sign in your binomial, you are done.
Solve the difference of squares. First make sure the numbers are arranged in the proper order, with the positive term before the negative term, then find the square root of each term. In the example above, the square root of y^2 is y, and the square root of 4 is 2.
Set up two sets of parentheses. Each will have the first square root followed by the second square root. In the first, they will be separated by a addition sign, and in the second, a subtraction sign. To take the example from steps 3 and 4, we get y^2 - 4 = (y + 2)(y - 2). Looking at the whole problem for step 3, we get 6y^2 - 24 = 6(y^2 - 4) = 6(y + 2)(y - 2).
Example : 7x -7A "7" can come out of each term, so Ill factor this out front:7x – 7 = 7( )Dividing the 7 out of "7x" leaves just an "x":7x – 7 = 7(x )What am I left with when I divide the 7 out of the second term? Well, if "nothing" is left, then "1" is left. (Remember: 7 ÷ 7 = 1.) So I get:7x – 7 = 7(x – 1) Take careful note: When "nothing" is left after factoring, a "1" is left behind in the parentheses.
Example : x2y3 + xyI can factor an "x" and a "y" out of each term: x2y3 = xy(x1y2) = xy(xy2) and xy = xy(1).x2y3 + xy = xy( )= xy(xy2 )= xy(xy2 + 1)
Example : 3x3 + 6x2 – 15x.I can factor a "3" and an "x" out of each term: 3x3 = 3x(x2), 6x2 = 3x(2x), and –15x = 3x(–5). Being careful of my signs, I get:3x3 + 6x2 – 15x = 3x( )= 3x(x2 )= 3x(x2 + 2x )= 3x(x2 + 2x – 5)