Find (if possible) the rational zeros of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x)=x^3-43x+42 x=square

To find the rational zeros of the polynomial function \( f(x) = x^3 - 43x + 42 \), we can use the Rational Root Theorem. This theorem states that any rational zero, expressed as a fraction \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (42) and \( q \) as a factor of the leading coefficient (1).<br /><br />The factors of 42 are:<br />\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42 \]<br /><br />Since the leading coefficient is 1, the possible rational zeros are simply the factors of 42:<br />\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42 \]<br /><br />We will test these possible zeros by substituting them into the polynomial \( f(x) \) to see if they yield zero.<br /><br />Let's start with \( x = 1 \):<br />\[ f(1) = 1^3 - 43(1) + 42 = 1 - 43 + 42 = 0 \]<br /><br />So, \( x = 1 \) is a zero.<br /><br />Next, let's try \( x = -1 \):<br />\[ f(-1) = (-1)^3 - 43(-1) + 42 = -1 + 43 + 42 = 84 \neq 0 \]<br /><br />Next, let's try \( x = 2 \):<br />\[ f(2) = 2^3 - 43(2) + 42 = 8 - 86 + 42 = -36 \neq 0 \]<br /><br />Next, let's try \( x = -2 \):<br />\[ f(-2) = (-2)^3 - 43(-2) + 42 = -8 + 86 + 42 = 120 \neq 0 \]<br /><br />Next, let's try \( x = 3 \):<br />\[ f(3) = 3^3 - 43(3) + 42 = 27 - 129 + 42 = -60 \neq 0 \]<br /><br />Next, let's try \( x = -3 \):<br />\[ f(-3) = (-3)^3 - 43(-3) + 42 = -27 + 129 + 42 = 144 \neq 0 \]<br /><br />Next, let's try \( x = 6 \):<br />\[ f(6) = 6^3 - 43(6) + 42 = 216 - 258 + 42 = 0 \]<br /><br />So, \( x = 6 \) is a zero.<br /><br />Next, let's try \( x = -6 \):<br />\[ f(-6) = (-6)^3 - 43(-6) + 42 = -216 + 258 + 42 = 84 \neq 0 \]<br /><br />Next, let's try \( x = 7 \):<br />\[ f(7) = 7^3 - 43(7) + 42 = 343 - 301 + 42 = 80 \neq 0 \]<br /><br />Next, let's try \( x = -7 \):<br />\[ f(-7) = (-7)^3 - 43(-7) + 42 = -343 + 301 + 42 = 0 \]<br /><br />So, \( x = -7 \) is a zero.<br /><br />Next, let's try \( x = 14 \):<br />\[ f(14) = 14^3 - 43(14) + 42 = 2744 - 602 + 42 = 2184 \neq 0 \]<br /><br />Next, let's try \( x = -14 \):<br />\[ f(-14) = (-14)^3 - 43(-14) + 42 = -2744 + 602 + 42 = -2100 \neq 0 \]<br /><br />Next, let's try \( x = 21 \):<br />\[ f(21) = 21^3 - 43(21) + 42 = 9261 - 903 + 42 = 8400 \neq 0 \]<br /><br />Next, let's try \( x = -21 \):<br />\[ f(-21) = (-21)^3 - 43(-21) + 42 = -9261 + 903 + 42 = -8216 \neq 0 \]<br /><br />Next, let's try \( x = 42 \):<br />\[ f(42) = 42^3 - 43(42) + 42 = 74088 - 1806 + 42 = 72224 \neq 0 \]<br /><br />Next, let's try \( x = -42 \):<br />\[ f(-