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Geometry A U6D3 HW
Factoring in Algebra
Factors
Numbers have factors:
And expressions (like x2+4x+3) also have factors:
Factoring
Factoring (called "Factorising" in the UK) is the process of finding the factors:
Factoring: Finding what to multiply together to get an expression.
It is like "splitting" an expression into a multiplication of simpler expressions.
1)What are ALL the factors of 12?
Example: factor 2y+6
Both 2y and 6 have a common factor of 2:
- 2y is 2 × y
- 6 is 2 × 3
So you can factor the whole expression into: 2y+6 = 2(y+3)
So 2y+6 has been "factored into" 2 and y+3
Factoring is also the opposite of Expanding:
2)Expand a) 5(x – 8)b) 2x( -x + 7)
3)Factor 10x – 15
Common Factor
In the previous example we saw that 2y and 6 had a common factor of 2
But to do the job properly make sure you have the highest common factor, including any variables
Example: factor 3y2+12y
Firstly, 3 and 12 have a common factor of 3.So you could have:
3y2+12y = 3(y2+4y)
But we can do better!3y2 and 12y also share the variable y.Together that makes 3y:
- 3y2 is 3y × y
- 12y is 3y × 4
So you can factor the whole expression into:
3y2+12y = 3y(y+4)
Check: 3y(y+4) = 3y × y + 3y × 4 = 3y2+12y
4)Factor 12x2+ 18x5) Factor 8xy2 – 24y3
More Complicated Factoring
Factoring Can Be Hard !
The examples have been simple so far, but factoring can be very tricky.
Because you have to figure what got multiplied to produce the expression you are given!
Experience HelpsBut the more experience you get, the easier it becomes.
Example: Factor 4x2 - 9
Hmmm... I can't see any common factors.
But if you know your Special Binomial Products you might see it as the "difference of squares":
Because 4x2 is (2x)2, and 9 is (3)2,
so we have:4x2 - 9 = (2x)2 - (3)2 And then we use the difference of squares formula:(a+b)(a-b) = a2 - b2
Where a is 2x, and b is 3. So let us try doing that:
(2x+3)(2x-3) = (2x)2 - (3)2 = 4x2 - 9
Yes!
So the factors of 4x2 - 9 are (2x+3) and (2x-3):Answer: 4x2 - 9 = (2x+3)(2x-3)
6) Factor 100x2 –817) Factor 121y2 – 49x2
Factoring Trinomials of the Form(Where the number in front of x squared is 1)
Basically, we are reversing the FOIL method to get our factored form. We are looking for two binomials that when you multiply them you get the given trinomial.
Step 1: Set up a product of two ( ) where each will hold two terms.
It will look like this: ( )( ).
Step 2: Find the factors that go in the first positions.
To get the x squared (which is the F in FOIL), we would have to have an x in the first positions in each ( ).
So it would look like this: (x )(x ).
Step 3: Find the factors that go in the last positions.
The factors that would go in the last position would have to be two expressions such that their product equals c (the constant) and at the same time their sum equals b (number in front of x term).As you are finding these factors, you have to consider the sign of the expressions:If c is positive, your factors are going to both have the same sign depending on b’s sign.If c is negative, your factors are going to have opposite signs depending on b’s sign.
Example 1: Factor the trinomial.
Note that this trinomial does not have a GCF.So we go right into factoring the trinomial of the form.
Step 1: Set up a product of two ( ) where each will hold two terms.It will look like this: ( )( )
Step 2: Find the factors that go in the first positions.Since we have y squared as our first term, we will need the following:(y )(y )
Step 3: Find the factors that go in the last positions.
We need two numbers whose product is 6 and sum is -5. That would have to be -2 and -3.Putting that into our factors we get: *-2 and -3 are two numbers whose prod. is 6and sum is -5
8) Factor x2+ 11x+189) Factor y2 – 13x + 12
Example 2: Factor the trinomialNote that this trinomial does have a GCF of 2y.
We need to factor out the GCF, as shown in Tutorial 27: The GCF and Factoring by Grouping, before we tackle the trinomial part of this.*Factor out the GCF of 2yWe are not finished, we can still factor the trinomial. It is of the form.Anytime you are factoring, you need to make sure that you factor everything that is factorable. Sometimes you end up having to do several steps of factoring before you are done.
Step 1 (trinomial): Set up a product of two ( ) where each will hold two terms.
It will look like this: 2y( )( )
Step 2 (trinomial): Find the factors that go in the first positions.
Since we have x squared as our first term, we will need the following: 2y(x )(x )
Step 3 (trinomial): Find the factors that go in the last positions.
We need two numbers whose product is -20 and sum is 1. That would have to be 5 and -4.
Putting that into our factors we get:*5 and -4 are two numbers whose prod. is -20and sum is 1
Note that if we would multiply this out, we would get the original trinomial.
10) Factor 11) Factor