Factoring a Difference of Cubes:
a3 – b3 = (a – b)(a2 + ab + b2)
This is equivalent to x^3 – 2^3. With the "minus" sign in the middle, this is a difference of cubes. To do the factoring, I'll be plugging x and 2 into the difference-of-cubes formula. Doing so, I get:
x^3 – 8 = x^3 – 23
= (x – 2)(x^2 + 2x + 2^2)
= (x – 2)(x^2 + 2x + 4)
Factoring a Sum of Cubes:
a3 + b3 = (a + b)(a2 – ab + b2)
The first term contains the cube of 3 and the cube of x. But what about the second term?
The 1 can be regarded as having been raised to any power, since 1 to any power is still just 1. so use the power of 3, since this will give you a sum of cubes. This means that the expression they've given can be expressed as:
(3x)^3 + 13
So, to factor, plug in 3x and 1 into the sum-of-cubes formula. This gives me:
27x^3 + 1 = (3x)^3 + 13
= (3x + 1)((3x)^2 – (3x)(1) + 1^2)
= (3x + 1)(9x^2 – 3x + 1)
Factoring Polynomials
Polynomials are algebraic expressions that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction, and multiplication.
Difference of two Squares
a2 – b2 = (a + b)(a – b)
A Binomial is a difference of two squares if both terms are squared and have different signs
Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
Step 2: Every difference of squares problem can be factored as follows:
a2 – b2 = (a + b)(a – b) or (a – b)(a + b). So, all you need to do to factor these types of problems is to determine what numbers squares will produce the desired results.
Step 3: Determine if the remaining factors can be factored any further.
Example:
X^2 - 16
(X + 4) (X - 4)
Verify answer by using FOIL
The first method for factoring polynomials will be factoring out the greatest common factor.
To use this method all that we do is look at all the terms and determine if there is a factor that is in common to all the terms. If there is, we will factor it out of the polynomial. This is using the distributive law in reverse.
Example:
8x^4 - 4x^3 + 10x^2
We first determine what is the greatest common factor we can take from the polynomial. We are able to take a 2x^2 from each of the terms in the polynomial. the factoring for the problem is:
2x^2 (4x^2 - 2x + 5)
To make sure its correct just factor it back out by using the distributive property
For grouping you would group the first two terms together and then the last two terms together. you then find the GCF between the two separate groups. than continue factoring out common factors until no factors are in common. This strategy may not always work.
Example:
x^5 + x - 2x^4 - 2
We would first group the first to terms and then the second two terms.
(x^5 + x) - (2x^4 - 2)
We will then can determine the greatest common factors between each group and pull it out of the terms.
In the first group of terms we can pull out an x
x(x^4 + 1)
The second group we can pull out a 2
-2(x^4 + 1)
we then put the two terms together
x(x^4 +1) -2(x^4 + 1)
So your factoring of the polynomial will be:
(x^4 + 1)(x - 2)
two make sure it is correct use the foil method and factor it back out to see if you get the original problem
Standard :
ax^2 + bx + c = 0
Find two numbers that multiply to get ac ( a times c), and add to get b
(a, b, and c can have any value, except that "a" can't be 0)
Example:
6x^2 + 5x - 6
So we want two numbers that multiply together to make -36, and add up to 5
now we determine the factors of 36.
1, 2, 3, 4, 6, 9, 12, 18, 36
We know that one of the numbers has to be a negative and the two numbers together must add up to 5
-4 & 9 would be the numbers we would need to use. so we would rewrite the 5x as:
6x^2 - 4x + 9x - 6
Now we will use the grouping property and group teh first two terms together than the second two terms
(6x^2 - 4x) + (9x - 6)
find the common factors of each
2x (3x -2) + 3 (3x-2)
the factor of the problem would be
(2x+3) (3x-2)