I need help finding all the zeros for one of my many polynomials:

x^5-2x^4+3x^3-20x^2+0x-3

Since the remainder is zero, then x = 4 is indeed a zero of –2x5 + 6x4 + 10x3 – 6x2 – 9x + 4, so:

Yes, x – 4 is a factor of –2x5 + 6x4 + 10x3 – 6x2 – 9x + 4

Find all the factors of 15x4 + x3 – 52x2 + 20x + 16 by using synthetic division.
Remember that, if x = a is a zero, then x – a is a factor. So use the Rational Roots Test (and maybe a quick graph) to find a good value to test for a zero (x-intercept). I'll try x = 1:

completed division

To find all the zeros of the polynomial x^5-2x^4+3x^3-20x^2+0x-3, you can use a technique called the Rational Root Theorem, followed by synthetic division. Here are the steps:

Step 1: Apply the Rational Root Theorem
The Rational Root Theorem helps us identify potential rational zeros (zeros in the form of p/q, where p divides evenly into the constant term and q divides evenly into the leading coefficient).

In our case, the constant term is -3 (the number without any x terms), and the leading coefficient is 1 (the coefficient of the highest power of x).

So, the possible rational zeros are of the form ± (factors of the constant term) / (factors of the leading coefficient).

In this case, the factors of -3 are 1, -1, 3, and -3, and the factors of 1 are 1 and -1. Thus, the possible rational zeros are ±1, ±3.

Step 2: Use Synthetic Division to Test the Possible Zeros
Synthetic division is a method used to evaluate polynomial functions and determine whether a given value is a zero.

You will have to perform synthetic division once for each possible rational zero.

Let's start with the first possible rational zero, which is 1:

1 │ 1 -2 3 -20 0 -3

Bring down the first coefficient (1) to the bottom row:
1 │ 1
│___

Multiply the divisor (1) by the last result (1), and write it below the next coefficient (-2):
1 │ 1 -2
│___
│ 1

Add the two numbers in the last column:
1 │ 1 -2
│___
│ 1 -1

Repeat the process, multiplying the divisor (1) by the new result (-1), and writing it below the next coefficient (3):
1 │ 1 -2 3
│___
│ 1 -1
│_____
2

Continue this process until you've reached the end of the polynomial:

1 │ 1 -2 3 -20 0 -3
│___ 1 -1 2 -18 -18
│_________________
1 -1 2 -18 -18 -21

The last number in the final row represents the remainder. If it is 0, it means that the divisor (in this case, 1) is a factor of the polynomial.

In our synthetic division, the remainder is -21, which is not 0. Therefore, 1 is not a zero of x^5-2x^4+3x^3-20x^2+0x-3.

Repeat the synthetic division process for the other possible rational zeros: -1, 3, and -3.

-1 │ 1 -2 3 -20 0 -3
│___ -1 3 -6 26 -6
│_________________
1 -3 6 -26 26 -9

For -1, the remainder is -9, so it is not a zero.

3 │ 1 -2 3 -20 0 -3
│___ 3 3 -18 -54 -162
│_________________
1 1 6 -38 -54 -165

For 3, the remainder is -165, so it is not a zero.

-3 │ 1 -2 3 -20 0 -3
│___ -3 -3 0 60 -180
│_________________
1 -5 0 40 60 -183

For -3, the remainder is -183, so it is not a zero.

None of the possible rational zeros ±1, ±3 are zeros of x^5-2x^4+3x^3-20x^2+0x-3.

Therefore, this polynomial does not have any rational zeros. If there are any zeros, they would be irrational or complex.