Let's solve the polynomial [tex]\( f(x) = x^3 - 7x^2 + 12x \)[/tex] by factoring it and then finding its rational zeros.
### Step 1: Factor the polynomial
First, observe if there is a common factor in all terms of the polynomial [tex]\( f(x) \)[/tex]. We can see that each term contains an [tex]\( x \)[/tex]. Therefore, we can factor out [tex]\( x \)[/tex].
[tex]\[ f(x) = x^3 - 7x^2 + 12x = x(x^2 - 7x + 12) \][/tex]
Next, we factor the quadratic expression [tex]\( x^2 - 7x + 12 \)[/tex] inside the parentheses.
To do so, we need to find two numbers that multiply to the constant term (12) and add up to the coefficient of the linear term (-7).
These numbers are -3 and -4, since:
[tex]\[ -3 \times -4 = 12 \][/tex]
[tex]\[ -3 + -4 = -7 \][/tex]
So we can factor [tex]\( x^2 - 7x + 12 \)[/tex] as [tex]\( (x - 3)(x - 4) \)[/tex].
Therefore, the polynomial [tex]\( f(x) \)[/tex] factors completely as:
[tex]\[ f(x) = x(x - 3)(x - 4) \][/tex]
### Step 2: Find the rational zeros
The rational zeros of the polynomial are the values of [tex]\( x \)[/tex] that make the polynomial equal to zero. Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x - 3 = 0 \Rightarrow x = 3 \)[/tex]
3. [tex]\( x - 4 = 0 \Rightarrow x = 4 \)[/tex]
So, the rational zeros of the polynomial [tex]\( f(x) = x^3 - 7x^2 + 12x \)[/tex] are:
[tex]\[ 0, 3, 4 \][/tex]
### Conclusion
The factored form of the polynomial is [tex]\( x(x - 3)(x - 4) \)[/tex] and the zeros are:
[tex]\[ \boxed{0}, \boxed{3}, \boxed{4} \][/tex]
