To find all roots of the equation [tex]\(x^3 - 5x^2 + 4x = 0\)[/tex], we need to solve for [tex]\(x\)[/tex] such that the given equation is satisfied. Let’s follow a step-by-step approach:
1. Factorize the equation: The given equation is a polynomial equation. We will start by factoring it.
[tex]\[
x^3 - 5x^2 + 4x = 0
\][/tex]
Notice that each term on the left-hand side contains [tex]\(x\)[/tex]. Hence, we can factor out [tex]\(x\)[/tex]:
[tex]\[
x(x^2 - 5x + 4) = 0
\][/tex]
2. Solve the factored equation: Now we have a product of factors equal to zero. To satisfy this equation, at least one of the factors must be zero. We solve for [tex]\(x\)[/tex] from each factor separately.
[tex]\[
x = 0
\][/tex]
and
[tex]\[
x^2 - 5x + 4 = 0
\][/tex]
3. Solve the quadratic equation: Next, we solve [tex]\(x^2 - 5x + 4 = 0\)[/tex]. This is a standard quadratic equation, and we can solve it using the factorization method.
First, we look for two numbers that multiply to give the constant term (4) and add up to give the coefficient of the linear term (-5). Those numbers are -1 and -4.
So, we can write:
[tex]\[
x^2 - 5x + 4 = (x - 1)(x - 4) = 0
\][/tex]
4. Find the roots of the quadratic equation: Setting each factor equal to zero gives us the roots:
[tex]\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\][/tex]
[tex]\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\][/tex]
5. Compile all the roots: Combining the roots from our factors, we get:
[tex]\[
x = 0, \quad x = 1, \quad x = 4
\][/tex]
Thus, all the roots of the equation [tex]\(x^3 - 5x^2 + 4x = 0\)[/tex] are:
[tex]\[
x = 0, \quad x = 1, \quad x = 4
\][/tex]
