Section 3.5 HW Score: 3/6 Answered: 4/6 Question 5 Find all zeros of f(x)=x^3-x^2-18x-10 Enter the zeros separated by commas. Enter exact value, not decimal approximations. square Question Help: El Video B Message instructor

To find the zeros of the polynomial function \( f(x) = x^3 - x^2 - 18x - 10 \), we need to solve for \( x \) such that \( f(x) = 0 \).<br /><br />First, we can use the Rational Root Theorem to identify possible rational roots. The Rational Root Theorem states that any rational root of the polynomial \( f(x) \) is a factor of the constant term divided by a factor of the leading coefficient.<br /><br />For \( f(x) = x^3 - x^2 - 18x - 10 \):<br />- The constant term is \(-10\).<br />- The leading coefficient is \(1\).<br /><br />The factors of \(-10\) are \(\pm 1, \pm 2, \pm 5, \pm 10\).<br /><br />We will 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 />\[ f(1) = 1^3 - 1^2 - 18 \cdot 1 - 10 = 1 - 1 - 18 - 10 = -28 \]<br />So, \( x = 1 \) is not a root.<br /><br />Next, let's try \( x = -1 \):<br />\[ f(-1) = (-1)^3 - (-1)^2 - 18 \cdot (-1) - 10 = -1 - 1 + 18 - 10 = 6 \]<br />So, \( x = -1 \) is not a root.<br /><br />Next, let's try \( x = 2 \):<br />\[ f(2) = 2^3 - 2^2 - 18 \cdot 2 - 10 = 8 - 4 - 36 - 10 = -42 \]<br />So, \( x = 2 \) is not a root.<br /><br />Next, let's try \( x = -2 \):<br />\[ f(-2) = (-2)^3 - (-2)^2 - 18 \cdot (-2) - 10 = -8 - 4 + 36 - 10 = 14 \]<br />So, \( x = -2 \) is not.<br /><br />Next, let's try \( x = 5 \):<br />\[ f(5) = 5^3 - 5^2 - 18 \cdot 5 - 10 = 125 - 25 - 90 - 10 = 0 \]<br />So, \( x = 5 \) is a root.<br /><br />Now that we have found one root, \( x = 5 \), we can use polynomial division or synthetic division to factor the polynomial and find the other roots.<br /><br />We can factor \( f(x) \) as:<br />\[ f(x) = (x - 5)(x^2 + 4x + 2) \]<br /><br />Now, we need to solve the quadratic equation \( x^2 + 4x + 2 = 0 \) using the quadratic formula:<br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br />where \( a = 1 \), \( b = 4 \), and \( c = 2 \).<br /><br />\[ x = \frac{-4 \pm \sqrt{16 - 8}}{2} \]<br />\[ x = \frac{-4 \pm \sqrt{8}}{2} \]<br />\[ x = \frac{-4 \pm 2\sqrt{2}}{2} \]<br />\[ x = -2 \pm \sqrt{2} \]<br /><br />So, the zeros of \( f(x) = x^3 - x^2 - 18x - 10 \) are:<br />\[ 5, -2 + \sqrt{2}, -2 - \sqrt{2} \]<br /><br />Therefore, the exact values of the zeros are:<br />\[ \boxed{5, -2 + \sqrt{2}, -2 - \sqrt{2}} \]