Question 4 Find all zeros of f(x)=x^3-10x^2+30x-24 Enter the zeros separated by commas. Enter exact value, not decimal approximations. square Question Help: Video (1) Message instructor

To find the zeros of the polynomial function \( f(x) = x^3 - 10x^2 + 30x - 24 \), we need to solve the equation \( f(x) = 0 \).<br /><br />First, we can try to find rational roots using the Rational Root Theorem, which states that any rational root of the polynomial \( f(x) \) is a factor of the constant term (-24) divided by a factor of the leading coefficient (1). Therefore, the possible rational roots are:<br />\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \]<br /><br />We can test these possible roots by substituting them into the polynomial to see if they yield zero.<br /><br />Let's start with \( x = 1 \):<br />\[ f(1) = 1^3 - 10(1)^2 + 30(1) - 24 = 1 - 10 + 30 - 24 = -3 \]<br />So, \( x = 1 \) is not a root.<br /><br />Next, let's try \( x = 2 \):<br />\[ f(2) = 2^3 - 10(2)^2 + 30(2) - 24 = 8 - 40 + 60 - 24 = 4 \]<br />So, \( x = 2 \) is not a root.<br /><br />Next, let's try \( x = 3 \):<br />\[ f(3) = 3^3 - 10(3)^2 + 30(3) - 24 = 27 - 90 + 90 - 24 = 3 \]<br />So, \( x = 3 \) is not a root.<br /><br />Next, let's try \( x = 4 \):<br />\[ f(4) = 4^3 - 10(4)^2 + 30(4) - 24 = 64 - 160 + 120 - 24 = 0 \]<br />So, \( x = 4 \) is a root.<br /><br />Now that we have found one root, we can use polynomial division to factor the polynomial and find the other roots. We divide \( f(x) \) by \( (x - 4) \):<br /><br />Using synthetic division or long division, we get:<br />\[ f(x) = (x - 4)(x^2 - 6x + 6) \]<br /><br />Now, we need to solve the quadratic equation \( x^2 - 6x + 6 = 0 \) to find the remaining roots. We can use the quadratic formula:<br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br />where \( a = 1 \), \( b = -6 \), and \( c = 6 \).<br /><br />Substituting these values into the formula, we get:<br />\[ x = \frac{6 \pm \sqrt{36 - 24}}{2} = \frac{6 \pm \sqrt{12}}{2} = \frac{6 \pm 2\sqrt{3}}{2} = 3 \pm \sqrt{3} \]<br /><br />So, the zeros of the polynomial \( f(x) = x^3 - 10x^2 + 30x - 24 \) are:<br />\[ 4, 3 + \sqrt{3}, 3 - \sqrt{3} \]