What happens when we want to change an equation from standard form into factored form?
There are several different methods that we can use to turn an equation from standard form into factored form. They are:
1. Common Factoring
Common factoring should always be the first thing you look for when you have an equation in standard form. You should look to see if each of the terms in the equation has a similar component that each of the terms can easily be divided into. The common factor can either be a variable (x) or a coefficient (any number.)
Example 1: Factor the expression 6x - 12.
You can notice that each term can be easily divided by the number 6.
6x - 12 = 6(x - 2)
Example 2: Factor the expression 7a² - 6a.
Now, each term has a common variable, a.
7a² - 6a = a (7a - 6)
Example 3: Factor the expression 3x² + 9x.
Now, each of the terms have two things in common: they have a common factor of 3, as well as a common factor of x.
3x² + 9x = 3x (x + 3)
2. Simple Trinomials
Simple trinomials have 3 terms, and can be written in the form y = ax² + bx + c, where x = 1. Examples of trinomials are y = x² + 2x + 1, y = x²- 6x + 9, and y = p² + 2p - 15.
When solving a simple trinomial, we must find two numbers that MULTIPLY to the last term, and two numbers that ADD to the middle term. One of the most common methods of solving a simple trinomial is called trial and error, in which we substitute different values into our work until we get the right answer.
Example 1: Evaluate y = x² + 2x + 1.
We must find a set of numbers that will multiply to give us 1(the last term), and that will add to give us 2(the middle term). What two numbers multiply to get the number 1?
Our options are 1 x 1, and -1 x -1.
Let's try 1 x 1.
y = x² + 2x + 1
y =(x + 1)(x + 1)
Now, let's use FOIL to see if we have used the right numbers.
y = x² + x + x + 1
y = x² + 2x + 1
Because our equation is the same as the one we were given, we have found the correct answer. If we would have used -1 and -1, we would have found x² - 2x + 1 to be our answer, which is NOT the same as the one we were given.
Example 2: Evaluate y = x² - 3x - 4.
Again, we must find factors that multiply to -4, and add to -3.
Factors of -4 are 2 x -2, 1 x -4 and -1 x 4.
Because we are using trial and error, it doesn't matter which set of factors we use first. Let's try 2 and -2.
y = (x + 2)(x - 2)
Using FOIL to check our solution, we get y = x² - 4, which is not the same expression as was given in the question.
We must try another set of factors. Let's try 1 and -4.
y = (x + 1)(x - 4)
Using FOIL to expand, we get y = x² - 3x - 4, which is the same as the expression we were originally given.
Thus, y = (x + 1)(x - 4) is the factored form of the expression y = x² - 3x - 4.
3. Difference of Squares
When working with a difference of squares, you must check to see that
1. Each term is a squared term and
2. There is a minus sign between the two terms.
When factoring a difference of squares, you put the square root of the first term at the beginning of each of the brackets, and the square root of the second term at the end of each of the brackets. The signs between them are positive in the first set of brackets, and negative in the second set of brackets.
It is very important that the signs are different in each of the brackets.
Example 1: Factor x² - 9.
First, check to make sure that each of the terms are perfect squares (x² and 3² in this case), and also check to make sure that the sign between the two is negative. Because this expression fulfills each of the two checks, we know that it is a difference of squares.
Now, let's factor.
x² - 9
=(x + 3)(x - 3)
Notice that the first term in each of the brackets is the square root of the first term, and the last term in each of the brackets is the square root of the last term, and that there are different signs in each of the brackets.
Example 2: Factor 4p² - 64q²
Again, the square root of the first term is 2p, and the square root of the second term is 8q. Also, there is a subtraction sign between them.
4p² - 64q² =(2p + 8q)(2p - 8q)
4. Complex Trinomials
Complex Trinomials follow the same rules as Simple Trinomials (see above), however the "a" term on a complex trinomial is not 1. That makes it a little trickier, and more often the method trial and error or guess and check will have to be used.
Examples of Complex Trinomials are 9n² - 6n + 1, 5x² + 17x + 6 and 2m² + 5m - 3. Notice that the "a" term does not equal 1 in each case.
Example 1: Factor 6x² + 13x - 5.
Again, we must look for factors of -5.
-5 x 1, and -1 x 5.
Also, we must find the factors of the first term (the "a" term).
1 x 6, 2 x 3 [If the "a" term was a negative, we would have to factor out a negative from each term in the trinomial. In other words, do not factor a complex trinomial until the "a" term is positive.]
Now we must randomly choose any two sets of numbers - we are performing guess and check.
Let's try -5 and 1 for the last term, and 1 and 6 for the first term.
(x - 5)(6x + 1)
=6x² - 29x - 5 ---> which is NOT what we want. Try new numbers.
Let's try 2 and 3 for the first term, and 5 and -1 for the second term.
(2x + 5)(3x - 1)
=6x² + 13x - 5 ---> which is what we want. This is the solution.
Example 2:Factor -3x² + 10x - 7.
First, we must rewrite the equation so that the "a" term is positive.
-3x² + 10x - 7
= - [3x² - 10x + 7] ---> notice that the negative is removed, and that brackets are put around the rest of the equation.
Now, find the factors of 3 (1 x 3); and the factors of 7 (7 x 1, and -7 x -1)
Let's use 1 x 3 for the first term (our only choice!), and -7 x -1 for the second term.
- 3x² + 10x - 7
= - [(3x - 7)(x - 1)]
If we were to expand the inside of the square brackets, we would get:
- [3x² - 10x + 7]
and if we were to expand the entire expression, we would get
- 3x² + 10x - 7 --->which is what we want.
Practice Questions
Page #307 #2(column 1), 3(column 2), 4(column 3), 6(odd), 8 a,c, 12.
