To factor the cubic polynomial [tex]\( f(x) = x^3 - 5x^2 + 4x \)[/tex] and find its rational zeros, we will follow these steps:
### Step 1: Factor the Polynomial
First, we look for common factors in the polynomial [tex]\( f(x) = x^3 - 5x^2 + 4x \)[/tex].
Notice that each term in the polynomial contains an [tex]\( x \)[/tex]. So, we can factor out an [tex]\( x \)[/tex] from the entire polynomial:
[tex]\[
f(x) = x(x^2 - 5x + 4)
\][/tex]
Next, we need to factor the quadratic part [tex]\( x^2 - 5x + 4 \)[/tex]. To do this, we look for two numbers that multiply to the constant term (4) and add up to the coefficient of the linear term (-5).
The two numbers that satisfy these conditions are -1 and -4 because:
[tex]\[
-1 \times -4 = 4
\][/tex]
[tex]\[
-1 + (-4) = -5
\][/tex]
Thus, the quadratic part can be factored as:
[tex]\[
x^2 - 5x + 4 = (x - 1)(x - 4)
\][/tex]
So, the complete factorization of the polynomial [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = x(x - 1)(x - 4)
\][/tex]
### Step 2: Find the Rational Zeros
The rational zeros of a polynomial are the values of [tex]\( x \)[/tex] that make the polynomial equal to zero. To find these zeros, we set each factor equal to zero:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x - 1 = 0 \)[/tex] \rightarrow [tex]\( x = 1 \)[/tex]
3. [tex]\( x - 4 = 0 \)[/tex] \rightarrow [tex]\( x = 4 \)[/tex]
### Step 3: Sort the Zeros from Least to Greatest
We list the zeros we found in ascending order:
[tex]\[
0, 1, 4
\][/tex]
### Final Answer:
Zeros: [tex]\( 0 \)[/tex], [tex]\( 1 \)[/tex], [tex]\( 4 \)[/tex]
