Answer:Therefore, the solutions to the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 are x = 0, x = 1, x = 5.
Explanation:
To solve the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0, we can use factoring by grouping.
First, we can group the first two and last two terms:
x^4 - 5x^3 + 7x^2 - 5x + 6 = (x^4 - 5x^3) + (7x^2 - 5x + 6)
Next, we can factor out x^3 from the first group and factor out 1 from the second group:
(x^3(x - 5)) + (7x^2 - 5x + 6)
Now, we can group the last two terms of the second group:
x^3(x - 5) + (7x^2 - 3x - 2x + 6)
Then, we can factor out 1 from the terms inside the parentheses and group them:
x^3(x - 5) + (7x^2 - 3x) + (-2x + 6)
Now, we can factor out x from the second and third groups:
x^3(x - 5) + x(7x - 3) - 2( x - 3)
We can simplify the third group by distributing the negative sign:
x^3(x - 5) + x(7x - 3) - 2x + 6
Finally, we can combine the second and third groups:
x^3(x - 5) + x(7x - 5) + 6
So, the factored form of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 is:
(x^3(x - 5) + x(7x - 5) + 6) = 0
This equation can be solved by setting each factor equal to zero and solving for x:
x^3(x - 5) + x(7x - 5) + 6 = 0
(x^3 - 7x^2 + 5x) + (6 - 5x) = 0
x(x^2 - 7x + 5) - (5x - 6) = 0
x(x - 5)(x - 1) - (5x - 6) = 0
x(x - 5)(x - 1) = 5x - 6
x^3 - 6x^2 + 10x - 6 = 5x - 6
x^3 - 6x^2 + 5x = 0
x(x^2 - 6x + 5) = 0
x(x - 1)(x - 5) = 0
Therefore, the solutions to the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 are x = 0, x = 1, x = 5.