use the rational zeros theorem to find all the real zeros of the polynomial function. use the zeros to factor f over the real numbers:

f(x)=x^4+2x^3-7x^2-8x+12

I am totally lost, please help?

all the rational possibilities are

±1,±2,±3,±4,±6,±12
That's a lot, but it's easy to check the small values. A little synthetic division shows that
f(x) = (x-1)(x^3+3x^2-4x-12)
= (x-1)(x-2)(x^2+5x+6)
= (x-1)(x-2)(x+2)(x+3)

Sure! I'd be happy to help you with finding the real zeros of the given polynomial function using the Rational Zeros Theorem and then factoring it over the real numbers.

The Rational Zeros Theorem states that if a polynomial function with integer coefficients has any rational zeros, then they must be in the form of p/q, where p is a factor of the constant term (in this case 12) and q is a factor of the leading coefficient (in this case 1).

Step 1: List all possible rational zeros
To find all possible rational zeros, we need to find the factors of the constant term 12 and the leading coefficient 1.
The factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12.
The factors of 1 (leading coefficient) are ±1.

Therefore, the possible rational zeros are all the combinations of the factors: ±1, ±2, ±3, ±4, ±6, ±12.

Step 2: Test the possible zeros using synthetic division
Now we will use synthetic division to quickly test each possible zero and find if it gives a remainder of zero.

Let's start with the first possible zero, say x = 1:
1 | 1 2 -7 -8 12
|_____
| 1 3 -4 -12
The remainder is 0, so x = 1 is a zero.

So, we have found a real zero x = 1. Now, let's factor the polynomial using this zero.

f(x) = (x - 1)(x^3 + 3x^2 - 4x - 12)

Now we have a new polynomial of degree 3: x^3 + 3x^2 - 4x - 12.

We can repeat steps 1 and 2 with this new polynomial to find other possible zeros. And if we find any, we can factor the polynomial further.

Using synthetic division again, we find that x = -3 is a zero. Therefore, we have:

f(x) = (x - 1)(x + 3)(x^2 + 6x - 4)

Now, we have a quadratic polynomial x^2 + 6x - 4, which can be factored further using factoring techniques or by using the quadratic formula.

By applying the quadratic formula, we find that the remaining factors are approximately:

x ≈ -4.5359 and x ≈ 0.5359

So, the factored form of f(x) over the real numbers is:

f(x) = (x - 1)(x + 3)(x + 4.5359)(x - 0.5359)

These are all the real zeros of the given polynomial function, and we have factored it over the real numbers.

I hope this helps! Let me know if you have any further questions.