How do you factor

y= -x^4 + x^3 + 3x^2 - 2x - 5

You may want to check the question.

The given polynomial has no real roots, and hence no rational/real factors.

See:
http://img62.imageshack.us/img62/1420/1285293974.png

I let f(x) = -x^4 + x^3 + 3x^2 - 2x - 5

I tried ±1, and ±5 in f(x) and none produced a zero,
so there are no rational linear factors.

To factor the given polynomial, we need to find its roots, which are the values of x that make y equal to zero. Once we find the roots, we can rewrite the polynomial as a product of linear factors.

Step 1: Check for rational roots
To begin, we can apply the rational root theorem to identify any potential rational roots of the polynomial. According to the theorem, the possible rational roots are the factors of the constant term (-5 in this case) divided by the factors of the leading coefficient (-1 in this case).

The factors of -5 are ±1 and ±5, while the factors of -1 are ±1. Therefore, the potential rational roots are ±1, ±5.

Step 2: Test potential rational roots
Now, we substitute each potential rational root into the polynomial equation and check if it equals zero.

Let's start with x = 1:
y = -1^4 + 1^3 + 3(1)^2 - 2(1) - 5
= -1 + 1 + 3 - 2 - 5
= -4

Since y ≠ 0 when x = 1, x = 1 is not a root.

Next, let's try x = -1:
y = -(-1)^4 + (-1)^3 + 3(-1)^2 - 2(-1) - 5
= -1 + (-1) + 3 - (-2) - 5
= -1 - 1 + 3 + 2 - 5
= -2

Again, y ≠ 0 when x = -1, so x = -1 is not a root either.

Now, let's check x = 5 and x = -5. However, these values are not likely to be roots as the polynomial coefficients are small, and large roots would lead to large values for y.

Since none of the potential rational roots worked, we need to use another technique to find the roots.

Step 3: Use numerical methods or a graphing calculator
We can use numerical methods like the Newton-Raphson method or the synthetic division method to find the roots. Alternatively, we can plot the graph of the polynomial using a graphing calculator or software to identify the x-intercepts, which correspond to the roots.

Without employing these methods, it is challenging to factor the given polynomial manually. However, once the roots are determined, say x = a, x = b, x = c, and x = d, the factored form of the polynomial can be written as:

y = (-x - a)(x - b)(x - c)(x - d)

Remember to substitute the actual values of the roots into the factored form.