Simplify and state the restriction 12x^4-25x+12/3x^2+2x-8

Question

mathstudent55 · Answer

Is this the problem? Are all the exponents of x correct?

$ \dfrac{12x^4 - 25x + 12}{3x^2 + 2x - 8} $

anonymous · Answer

Actually its 12x^2 instead of 12x^4

mathstudent55 · Answer

That makes more sense.

mathstudent55 · Answer

$\dfrac{12x^2 - 25x + 12}{3x^2 + 2x - 8}$

The first step is to factor the numerator and denominator. Can you do that?

anonymous · Answer

No i dont remember

mathstudent55 · Answer

Ok. Let's start by factoring the numerator.
You need to factor a trinomial of the form
$ax^2 + bx + c$

mathstudent55 · Answer

We will factor by grouping. Here are the steps.
1. Multiply ac together.
Your trinomial is $12x^2 - 25x + 12$, so
$a = 12$, $b = -25$, and $c = 12$
What is ac for your trinomial?

anonymous · Answer

144

mathstudent55 · Answer

Good.
2. Find two factors of ac that add to b.

In your case, find two factors of 144 that add to -25.
Notice that 144 is positive, two factors of 144 are either both positive or both negative. Since b is negative, you need two negative factors of 144.

anonymous · Answer

Is it -12 and -12

mathstudent55 · Answer

-12 * (-12) = 144
-12 + (-12) = -24
The two factors do multiply to 144, but they add to -24, not -25. So they don't work. You need two different factors of 144.

anonymous · Answer

I cant find anything that adds up to -25 and equals 144 :/

mathstudent55 · Answer

I can just give you the numbers, but instead let's work on it togehter.
One method that is helpful in this is to find all the prime factors of the number.
Do you know how to find prime the factors of a number?

mathstudent55 · Answer

The prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.

anonymous · Answer

So what do i do with these?

mathstudent55 · Answer

Divide the number by the smallest prime number as many times as possible. When it is no longer divisible by that prime, move on to the next prime number. Keep on dividing until your division results in a 1. Then all the numbers you divided by are the prime factors of the number you started with. 

This is how you get the prime factors of 144:
144/2 = 72
72/2 = 36
36/2 = 18
18/2 = 9
9/3 = 3
3/3 = 1

mathstudent55 · Answer

This means that 144 = 2 * 2 * 2 * 2 * 3 * 3

anonymous · Answer

ohh

mathstudent55 · Answer

Now group prime factors together into two factors:
(2 * 2 * 2 * 2 * 3) * (3) = 48 * 3 = 144, but 48 + 3 = 51 (not what we need)

Try:
(2 * 2 * 2 * 2) * (3 * 3) = 16 * 9 = 144 and 16 + 9 = 25.
So pick -16 and -9: -16 * (-9) = 144 and -16 + (-9 ) = -25 (just what we need)

mathstudent55 · Answer

Now we are ready for step 3 in the factoring:
3. Break up the middle term bx into two terms using these 2 factors of ac.

In your problem, rewrite -25x as -16x - 9x, so now you need to factor:
$12x^2 - 16x - 9x + 12$

mathstudent55 · Answer

For factoring a 4-term polynomial, we use the factoring by grouping method.

mathstudent55 · Answer

4. Factor the 4-term polynomial by grouping.

This means, factor a common term out of the first two terms, and factor a common term out of the last two terms.

For your problem, this is:

$ 4x(3x - 4) - 3(3x - 4) $

Now factor out the common term of $3x - 4$:

$ (3x - 4)(4x - 3) $

Now the numerator is factored.

mathstudent55 · Answer

Now you need to factor the denominator using the same method.

anonymous · Answer

So i use the same method?

mathstudent55 · Answer

Yes.

mathstudent55 · Answer

Follow the numbered steps and try it. If you need help, let me know.

mathstudent55 · Answer

To factor $ ax^2 + bx + c $, factor by grouping:
1. Multiply ac together.
2. Find two factors of ac that add to b.
3. Break up the middle term bx into two terms using these 2 factors of ac.
4. Factor the 4-term polynomial by grouping.