solve the equation in the real number system.

2x^4-23x^3+75x^2-88x+28=0
What are the real solutions of the equations?

please show work. I do not know how to work this at all

x f(x)

1 -6
0 28
-1 216

so there is a zero between zero x=0 AND x = 1
You can iterate and find it
then divide by (x-xo)
However before doing that I would search on Google for a polynomial grapher.

http://mathportal.org/calculators/polynomials-solvers/polynomial-graphing-calculator.php

yields 1/2 , 2 , 7

by the way you can see from the graph that 2 is a double root

If you were clever enough to factor this you would have
(2x-1)(x-2)(x-2)(x-7)

To solve the given equation 2x^4 - 23x^3 + 75x^2 - 88x + 28 = 0, we can use the method of factoring, followed by the zero-product property.

Step 1: Factor out common terms (if any):
First, let's see if there are any common factors among the terms. In this case, we can't factor out any common factors.

Step 2: Look for rational roots:
Now, let's find the possible rational roots of the equation using the Rational Root Theorem. According to the theorem, the possible rational roots can be expressed as p/q, where p is a factor of the constant term 28, and q is a factor of the leading coefficient 2:

p = ±1, ±2, ±4, ±7, ±14, ±28 (possible factors of 28)
q = ±1, ±2 (possible factors of 2)

The possible rational roots are: ±1, ±2, ±4, ±7, ±14, ±28.

Step 3: Test the possible roots using synthetic division:
We will use synthetic division to test each possible rational root found in Step 2. We want to find a root, if any, that gives us a zero remainder when substituted into the equation.

Let's start by testing the root x = 1:

2 | 2 -23 75 -88 28
| 2 -21 54 -34
|___________________
2 -21 54 -34 -6

The remainder is -6, which means x = 1 is not a root.

Next, let's test the root x = -1:

2 | 2 -23 75 -88 28
| -2 25 -100 188
|___________________
2 -25 50 -188 216

The remainder is 216, so x = -1 is not a root either.

Continuing this process, we test the remaining possible roots, and none of them yields a zero remainder. Therefore, the equation has no rational roots.

Step 4: Use numerical methods:
Since we have determined that there are no rational roots, we can resort to numerical methods to find approximate solutions. One common method is using a graphing calculator or a software program that can plot the graph of the equation. By analyzing the graph, we can find the approximate real solutions.