Question

factor F(x) =x^4+x^3-8x^2+6x+36 completey.

please show me all the steps to do this. This is something i havnt been able to do and its a big part of the section im currently in in school.

Answer 1

If you have a zero of the polynomial (call it a), then you can write it as:

(x-a)(3rd degree polynomial).

In these kind of problems, there are often a few "easy" zeroes, like 1, -1, 2, -2, 3, -3, etc.
So try a few of these, you will find a zero!

To get the 3rd degree polynomial, you need to do a synthetic division.

Does this ring a bell, or are you still in the dark?

Answer 2

im still very confused

Answer 3

First try a few of the numbers: 1, -1, 2, -2, 3, -3.
So: set x = 1 and see what F(1) is (nonzero in this case)
Then calculate F(-1), F(2), F(-2)....

Answer 4

how do i even do that? do i multiply what i put in the place of x by all the other numbers? like F(1)= 1^4 instead of x^4?

Answer 5

Replace every x with 1:\[F(x) =1^4+1^3-8 \cdot 1^2+6\cdot1+36 \]

Answer 6

what am i looking to accomplish with substituting the numbers for x? how do i know which one is correct?

Answer 7

You are looking for zeroes. Mostly, these functions are made so that there are nice numbers like 1 or -2 that make it zero.
What does that have to do with factoring?
Well, the zeroes are the factors you are looking for, in a way:

Let us reverse the process:

Say you have (x+2)(x+3).
If you set x=-2, you get (-2+2)(-2+3)=0*1=0, so -2 makes it zero.
Also -3 makes it zero.
Whenever a factor, be it x+2 or x+3 is zero, the whole thing is zero.

Now you have x^4+x^3-8x^2+6x+36.
This is a rather complex polynomial. If only you had a zero of it...
I'll tell you a zero: it is -2. This means you can write
x^4+x^3-8x^2+6x+36 as

(x+2)(3rd degree polynomial)

The only problem you have now is: how to find the 3rd degree polyomial?

Answer 8

do you find the third degree by what you were telling me to do before?

Answer 9

You can find the 3rd degree one by dividing x^4+x^3-8x^2+6x+36 by x+2.
A synthetic division is quick and easy, if you have done it before.

If goes as follows:

write the zero you found: -2 along with the coefficients of F:

-2 1 1 -8 6 36

Then do the following:

-2 1 1 -8 6 36

+ ___________________________
1

The first coefficient is dropped below the line.
Then multiply -2 with that 1, put the result in the next column and add:

-2 1 1 -8 6 36
-2
+ ___________________________
1 -1

Repeat this process:
Multiply -2 with -1, put in the next column and add:

-2 1 1 -8 6 36
-2 2
+ ___________________________
1 -1 -6

Keep on doing this:

-2 1 1 -8 6 36
-2 2 12
+ ___________________________
1 -1 -6 18

Last time:

-2 1 1 -8 6 36
-2 2 12 -36
+ ___________________________
1 -1 -6 18 0

The 0 indicates that the division has no rest term. We have found:

F(x) = (x + 2)(x³ - x² - 6x + 18)

So we have partly factored F(x).

Answer 10

oh my goodness. This is a lot more then i thought i had to do. Thats only partly factored now?

Answer 11

Yes, because there is another zero: -3.
If you calculate F(-3)=(-3 + 2)((-3)³ - (-3)² - 6(-3) + 18)=(-1)(-27-9+18+18)=(-1)*0=0.
So now we know:

F(x)=(x+2)(x+3)(2nd degree polynomial).

We have now the same problem as earlier: if we do a synthetic division of x³ - x² - 6x + 18 by x+3, we will know the 2nd degree polynomial:

Write -3 and coefficients of x³ - x² - 6x + 18 on a line, and drop the first coefficient below:

-3 1 -1 -6 18

+----------------
1

Multiply and add:

-3 1 -1 -6 18
-3 12 -18
+-----------------
1 -4 6 0

So the second degree polynomial is x²-4x+6, and our function becomes:

F(x) = (x + 2)(x + 3)(x² - 4x + 6)

Answer 12

Now we have two possibilities:

1. You haven't done complex numbers yet
2. You know about complex numbers.

If 1. is the case, you can check that there are no more zeroes, because if you look at
x² - 4x + 6, you will notice that b²-4ac = (-4)² - 4*1*6=16 - 24 = -8 < 0, so the discriminant is <0, which means the equation x² - 4x + 6 = 0 has no (real) zeroes.
There are no more zeroes, so you cannot factor any further.

If 2. is the case, you can use the Quadratic Formula to calculate the two complex zeroes and factor a bit more.

Let me know if it is 1. or 2.!

Answer 13

i dont know about complex numbers

Answer 14

You are a lucky girl, because you're done.

The completely factored version of the function is:
F(x) = (x + 2)(x + 3)(x² - 4x + 6).

Answer 15

thank you so much for explaining this to me. ive been having such a hard time with these and its causing me to fail. so thank you for all your help