To find the zeros of the polynomial function [tex]\( f(x) = x^3 - 10x^2 + 24x \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. Here are the detailed steps to find the zeros:
1. Factor the Polynomial:
Let's start by factoring the polynomial [tex]\( f(x) \)[/tex].
[tex]\[ f(x) = x^3 - 10x^2 + 24x \][/tex]
Notice that we can factor out an [tex]\( x \)[/tex] from each term:
[tex]\[ f(x) = x(x^2 - 10x + 24) \][/tex]
So, we have factored the polynomial into two parts: [tex]\( x \)[/tex] and [tex]\( x^2 - 10x + 24 \)[/tex].
2. Find the Zeros of Each Factor:
First, let's find the zero from the factor [tex]\( x \)[/tex].
[tex]\[ x = 0 \][/tex]
So, one zero of the polynomial is [tex]\( x = 0 \)[/tex].
Now, let's find the zeros of the quadratic polynomial [tex]\( x^2 - 10x + 24 \)[/tex].
We can solve the quadratic equation [tex]\( x^2 - 10x + 24 = 0 \)[/tex] either by factoring, using the quadratic formula, or completing the square. Let's use factoring.
[tex]\[ x^2 - 10x + 24 = (x - 4)(x - 6) \][/tex]
So, the quadratic polynomial can be factored as [tex]\( (x - 4)(x - 6) \)[/tex].
3. Solve for the Remaining Zeros:
[tex]\[ (x - 4) = 0 \quad \text{or} \quad (x - 6) = 0 \][/tex]
Setting each factor equal to zero, we find the additional zeros:
[tex]\[ x - 4 = 0 \Rightarrow x = 4 \][/tex]
[tex]\[ x - 6 = 0 \Rightarrow x = 6 \][/tex]
4. Conclusively List All Zeros:
Therefore, the zeros of the polynomial function [tex]\( f(x) = x^3 - 10x^2 + 24x \)[/tex] are:
[tex]\[ x = 0, \, x = 4, \, x = 6 \][/tex]
So, the correct answer is:
[tex]\[ 6, 0, 4 \][/tex]
Therefore, the answer is:
rect [tex]${data-answer}gt;$[/tex] [tex]$6,0,4$[/tex]
