Question

Which of the following lists all of the roots of

X^5 - 3x^4 + 3x^3 - x^2 =0

Why is the answer ( 0, 1)

your answer was this:
x^5 - 3x^4 + 3x^3 - x^2
= x^2(x^3-3x^2+3x-1)
= x^2(x-1)^3

but i dont get the last part..(= x^2(x-1)^3)

Answer 1

Recall Pascal's Triangle

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...
The nth row is the coefficients of (a+b)^n
As soon as I saw 1 -3 3 -1 I knew.
Failing that, you could try plugging some values, knowing that all the roots would be +1 or -1 (See Rational Roots Theorem)

Answer 2

[AI-generated answer — Step-by-Step Bot]

To understand why the list of roots is (0, 1), let me explain the steps in more detail:

1. Begin with the polynomial equation: x^5 - 3x^4 + 3x^3 - x^2 = 0.

2. Factor out x^2 from the equation: x^2(x^3 - 3x^2 + 3x - 1) = 0.

3. Now, consider the expression in parentheses: x^3 - 3x^2 + 3x - 1. We want to find the values of x that make this expression equal to zero.

4. If we factor out (x - 1) from the expression x^3 - 3x^2 + 3x - 1, we get: (x - 1)(x^2 - 2x + 1).

5. Simplifying further, (x - 1)(x - 1)(x - 1) = (x - 1)^3.

6. Therefore, the equation x^5 - 3x^4 + 3x^3 - x^2 = 0 can be written as x^2(x - 1)^3 = 0.

By setting each factor equal to zero, we can determine the roots:

x^2 = 0, which gives x = 0.

(x - 1)^3 = 0, which gives x = 1 (since (x - 1)^3 = 0 when (x - 1) = 0).

So, the complete list of roots is (0, 1).

Answer 3

[AI-generated answer — Explain Bot]

To find the roots of the equation X^5 - 3x^4 + 3x^3 - x^2 = 0, we can factor out common terms from the equation.

So, let's factor out x^2 from the equation:
x^5 - 3x^4 + 3x^3 - x^2 = x^2(x^3 - 3x^2 + 3x - 1)

Now, we have the equation x^2 multiplied by another expression (x^3 - 3x^2 + 3x - 1).

To find the roots of this equation, we need to set both x^2 and (x^3 - 3x^2 + 3x - 1) equal to zero.

1. x^2 = 0
From this equation, we can see that x = 0 is a root.

2. (x^3 - 3x^2 + 3x - 1) = 0
This is a cubic equation, and it can be difficult to find its roots. However, we can observe that (x - 1) is a factor, which we can confirm by substituting x = 1 into the equation:

(1^3 - 3(1)^2 + 3(1) - 1) = 0
(1 - 3 + 3 - 1) = 0
0 = 0

Since substituting x = 1 yields a true statement, we can conclude that (x - 1) is a factor.

Now, we can apply synthetic division or long division to further factorize the cubic expression:

(x^3 - 3x^2 + 3x - 1) divided by (x - 1)

1 | 1 - 3 + 3 - 1
| 1 - 2 + 1
________________
1 - 2 + 1 + 0

The result is x^2 - 2x + 1, which can be factored as (x - 1)(x - 1).

So, the two remaining roots are x = 1.

Combining all the roots we found:
x = 0, x = 1

Therefore, the complete list of roots for the equation X^5 - 3x^4 + 3x^3 - x^2 = 0 is (0, 1).