Hey how would you factor this polynomial to find the zeroes of this function?

x^4-5x^3+7x^2+3x-10

Always try x = 1 and x = -1

If x = 1 the function is -4
but if x = -1 the function is zero.
Therefore (x+1) is a factor, so use long division.
(x+1)(x^3-6x^2+13x-10)
Now we need something that works with 10 like 10 and 1 or 5 and 2
Try 2 and -2
2 works so (x-2) is a factor. Divide
(x^3-6x^2+13x-10) by (x-2)
(x+1)(x-2)(x^2-4x+5)
the last one you must do by quadratic equation and the roots are complex
x = (2+i) and x = (2-i)
so
(x+1)(x-2)(x-2+i)(x-2-i)
and
x = -1, 2 , 2+i, 2-i

Try +/- 1,2,5 and 10, the integer factors of 10. That will apply the "rational roots" theorem.

You will see right away that x = -1 makes the polynomial zero. Therefore x+1 is a factor.

Divide x^4-5x^3+7x^2+3x-10 by x+1 for the other factor, a cubic polynomial.
That cubic factor is
x^3 -6x^2 +13x -10
x=2 makes that zero, so x-2 is a factor.
Divide x^3 -6x^2 +13x -10 by x-2 to get a quadratic factor. It will be
x^2 -4x + 5
Then see if you can factor that. The remaining roots are complex.

To factor the polynomial and find the zeroes of the function, you can use a method called polynomial factoring. Here's how you can do it:

Step 1: Check for any common factors
First, check if there are any common factors among the coefficients of the polynomial. In this case, there are no common factors.

Step 2: Use the Rational Root Theorem
Apply the Rational Root Theorem to find possible rational roots or zeroes of the polynomial. According to the theorem, any rational root of the polynomial must be in the form of p/q, where p is a factor of the constant term (in this case, -10) and q is a factor of the leading coefficient (in this case, 1).

To find the factors of -10, you can list all the possible combinations: ±1, ±2, ±5, ±10. To find the factors of 1, you list the possibilities as ±1.

Therefore, the possible rational roots are ±1, ±2, ±5, ±10.

Step 3: Test the possible roots using synthetic division
Using synthetic division, test each possible root to see if it produces a zero remainder. Divide the polynomial by each possible root one by one until you find a root that results in a zero remainder.

For example, let's test the root x = 1:
Perform synthetic division:
1 | 1 -5 7 3 -10
1 -4 3 6 -4
The remainder is -4, which means that (x - 1) is not a factor of the polynomial.

Continue trying different possible roots until you find one that produces a zero remainder. In this case, after trying a few more roots, you'll find that x = 2 is a root that produces a zero remainder.

Step 4: Factor the polynomial
Once you find one root, you can factor the polynomial by dividing it by (x - root). In this case, divide the polynomial by (x - 2):

(x - 2) (x^3 - 3x^2 + 13x + 5)

Now, you have a reduced polynomial of degree 3.

Step 5: Repeat the process
Repeat the process by checking for common factors and using the Rational Root Theorem on the reduced polynomial. You can continue factoring until you cannot factor further.

In this case, the reduced polynomial x^3 - 3x^2 + 13x + 5 can be factored using numerical or graphical methods, and one of the factors is (x - 2). However, the remaining factors are not easily factorable using integers. Therefore, the polynomial x^4 - 5x^3 + 7x^2 + 3x - 10 has one real root at x = 2, and the other roots are either irrational or complex numbers.