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) = 4x3 − 34x2 + 80x − 32

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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) = 4x3 − 34x2 + 80x − 32

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Apply Rational Root Theorem: To find the rational zeros of the polynomial, we can use the Rational Root Theorem, which states that any rational zero, expressed in its lowest terms p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.List Factors: First, we list the factors of the constant term (-32) and the leading coefficient (4). Factors of -32: ±1, ±2, ±4, ±8, ±16, ±32 Factors of 4: ±1, ±2, ±4Form Possible Fractions: Now, we form all possible fractions p/q using the factors of -32 for p and the factors of 4 for q. We simplify them if necessary to get all possible rational zeros. Possible rational zeros: ±1, ±1/2, ±1/4, ±2, ±2/4 (which simplifies to ±1/2), ±4, ±4/4 (which simplifies to ±1), ±8, ±8/4 (which simplifies to ±2), ±16, ±16/4 (which simplifies to ±4), ±32, ±32/4 (which simplifies to ±8)Test Rational Zeros: We test each possible rational zero by substituting it into the polynomial f(x) and checking if it results in zero. This can be done using synthetic division or direct substitution.Test x = 1: Let's start by testing x = 1: f(1) = 4(1)^3 − 34(1)^2 + 80(1) − 32 = 4 − 34 + 80 − 32 = 18 Since f(1) ≠ 0, x = 1 is not a zero.Test x = -1: Next, we test x = -1: f(-1) = 4(-1)^3 − 34(-1)^2 + 80(-1) − 32 = -4 − 34 − 80 − 32 = -150 Since f(-1) ≠ 0, x = -1 is not a zero.Test Remaining Zeros: We continue testing the remaining possible rational zeros. After testing all of them, we find that x = 2 is a zero: f(2) = 4(2)^3 − 34(2)^2 + 80(2) − 32 = 32 − 136 + 160 − 32 = 24 This is incorrect, as f(2) ≠ 0, so x = 2 is not a zero.

Find (if possible) the rational zeros of the function. $\newline$ (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) $\newline$ $f(x) = 4x^3 - 34x^2 + 80x - 32$

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Find (if possible) the rational zeros of the function. \newline(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) \newlinef(x)=4x3−34x2+80x−32f(x) = 4x^3 - 34x^2 + 80x - 32f(x)=4x3−34x2+80x−32

Find (if possible) the rational zeros of the function. $\newline$ (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) $\newline$ $f(x) = 4x^3 - 34x^2 + 80x - 32$