what are the zeros for 1x^5+2x^4+4x^3+19x^2+9x^2+9x-5?
5th-degree polynomials are not open to solution in general, so look for low-hanging fruit using synthetic division.
any rational roots must be ±1 or ±5.
no joy there.
not sure how to find the roots analytically. You may have to use some iterative methods. Don't know whether you have studied those yet.
If you go to wolframalpha.com and enter
solve x^5+2x^4+4x^3+19x^2+9x^2+9x-5 = 0
you will get the roots, but not much help in figuring out the exact values using radicals.
To find the zeros of the polynomial 1x^5 + 2x^4 + 4x^3 + 19x^2 + 9x^2 + 9x - 5, we can use a method called factoring or polynomial long division. However, in this case, the polynomial doesn't appear to have an easily factorable form, so let's try using synthetic division to find the zeros.
Synthetic division allows us to evaluate the polynomial at a given value to check if it is a zero. We will check integers first, starting with -5, -4, -3, and so on, until we find a value that yields 0.
Let's begin by trying the value x = -1 using synthetic division:
-1 | 1 2 4 19 9 -5
---------------------
-1 -1 -3 -16 7
---------------------
1 1 3 3 2
The result of this division is a new polynomial: 1x^4 + 1x^3 + 3x^2 + 3x + 2.
Now, let's try x = -2:
-2 | 1 1 3 3 2
-------------------
-2 2 -2 -2
--------------
1 -1 1 1
The result this time is: 1x^3 - 1x^2 + 1x + 1.
Next, let's try x = -3:
-3 | 1 -1 1 1
------------
-3 12 -39
-----------
1 9 -38
The result is: 1x^2 + 9x - 38.
Now, let's try x = -4:
-4 | 1 9 -38
----------
-4 -20 112
-----------
1 -11 74
Finally, let's try x = -5:
-5 | 1 -11 74
-----------
-5 80 -70
-----------
1 15 4
After performing synthetic division for each test value, we have obtained a quadratic equation: 1x^2 + 15x + 4. This equation can be factored as (x + 1)(x + 4), resulting in the factors: (x + 1)(x + 4)(x^2 + 9x - 38).
Therefore, the zeros (or roots) of the original polynomial 1x^5 + 2x^4 + 4x^3 + 19x^2 + 9x^2 + 9x - 5 are: x = -1, x = -4, and the solutions to x^2 + 9x - 38 = 0. You can use methods like factoring, completing the square, or quadratic formula to solve the quadratic equation and find the remaining zeros.