how can i factor this out completly
x^4+x^3-12
i know that4*-3=-12 but how can i get that middle term
x^2*x^2=x^4
first you cannot take out any GCF.
so if you decide to use grouping, you can create an equivalent problem:
x^4 + x^3 - 6 - 6
then by using grouping (the x-terms and the integers):
x^3 ( x + 1) - 6(1+1)
simplified, that is:
(x^3 - 6) - (x + 3) [combining the parentheses)
you can now simply factor out the x-cubed binomial with less difficulty.
as you could *probably* see, you cannot factor this, so using the quadratic formula:
x^3 + 0x -6
a = 1
b = 0
c = -6
-b +/- sqrt(b^2 - 4ac) / 2a
0 +/- sqrt(0 - 24) / 2
(since sqrt(-24) doesn't exist, you have to use i)
0 +/- 2 x 6i / 2
a) 0 + 2 x 6i / 2
12 i / 2
6 i
b) 0 -2 x 6i / 2
-12 i / 2
-6 i
so x could equal 6i or -6i
Sorry if this was tedious, but it wasn't easy. hope you understand.
To factor the expression x^4 + x^3 - 12 completely, let's break down the process step by step:
1. Start with the expression: x^4 + x^3 - 12
2. First, check if there is any common factor (Greatest Common Factor, GCF) among the terms. In this case, there isn't a common factor other than 1.
3. Next, let's see if we can apply the grouping method to factor the expression. By grouping, we can rearrange the terms to find a common factor.
We can rewrite the expression as: x^4 + x^3 - 6x - 6x - 12
Now, we can group the terms: (x^4 + x^3) - (6x + 6x) - 12
4. Simplifying the groups, we get: x^3(x + 1) - 6(x + 1) - 12
5. Now, we can see that we have a common binomial factor of (x + 1). We can factor it out: (x + 1)(x^3 - 6) - 12
6. Now, let's focus on factoring further the remaining binomial (x^3 - 6). Unfortunately, this binomial cannot be factored nicely since it doesn't possess any simple factors that would allow easy factoring.
7. As a result, we have factored the expression as: (x + 1)(x^3 - 6) - 12
Therefore, the expression x^4 + x^3 - 12 cannot be factored completely into simpler terms.