Zeros of Polynomial

Factors of the leading term 12 are ±1, ±2, ±3, ±4, ±6, ±12

Factors of the constant term -2 are ±1, ±2

Possible Rational Zeros: ±1/1, ±1/2, ±1/3, ±1/4, ±1/6, ±1/12, ±2/1, ±2/2, ±2/3, ±2/4, ±2/6, ±2/12

Possible Rational Zeros (reduced): ±1, ±1/2, ±1/3, ±1/4, ±1/6, ±1/12, ±2, ±1, ±2/3, ±1/2, ±1/3, ±1/6

Now we will check each one of these possible rational zeros to see which one is a zero of the function.

f(x) = 12x³ + 14x² - 3x - 2

f(1) = 12(1)³ + 14(1)² - 3(1) - 2 = 21

f(-1) = 12(-1)³ + 14(-1)² - 3(-1) - 2 = 3

f(1/2) = 12(1/2)³ + 14(1/2)² - 3(1/2) - 2 = 3/2

f(-1/2) = 12(-1/2)³ + 14(-1/2)² - 3(-1/2) - 2 = 3/2

f(1/3) = 12(1/3)³ + 14(1/3)² - 3(1/3) - 2 = -1

f(-1/3) = 12(-1/3)³ + 14(-1/3)² - 3(-1/3) - 2 = 1/9

f(1/4) = 12(1/4)³ + 14(1/4)² - 3(1/4) - 2 = -27/16

f(-1/4) = 12(-1/4)³ + 14(-1/4)² - 3(-1/4) - 2 = -9/16

f(1/6) = 12(1/6)³ + 14(1/6)² - 3(1/6) - 2 = -37/18

f(-1/6) = 12(-1/6)³ + 14(-1/6)² - 3(-1/6) - 2 = -7/6

f(1/12) = 12(1/12)³ + 14(1/12)² - 3(1/12) - 2 = -103/48

f(-1/12) = 12(-1/12)³ + 14(-1/12)² - 3(-1/12) - 2 = -239/144

f(2) = 12(2)³ + 14(2)² - 3(2) - 2 = 144

f(-2) = 12(-2)³ + 14(-2)² - 3(-2) - 2 = -36

f(1) = 12(1)³ + 14(1)² - 3(1) - 2 = 21

f(-1) = 12(-1)³ + 14(-1)² - 3(-1) - 2 = 3

f(2/3) = 12(2/3)³ + 14(2/3)² - 3(2/3) - 2 = 52/9

f(-2/3) = 12(-2/3)³ + 14(-2/3)² - 3(-2/3) - 2 = 8/3

f(1/2) = 12(1/2)³ + 14(1/2)² - 3(1/2) - 2 = 3/2

f(-1/2) = 12(-1/2)³ + 14(-1/2)² - 3(-1/2) - 2 = 3/2

f(1/3) = 12(1/3)³ + 14(1/3)² - 3(1/3) - 2 = -1

f(-1/3) = 12(-1/3)³ + 14(-1/3)² - 3(-1/3) - 2 = 1/9

f(1/6) = 12(1/6)³ + 14(1/6)² - 3(1/6) - 2 = -37/18

f(-1/6) = 12(-1/6)³ + 14(-1/6)² - 3(-1/6) - 2 = -7/6

There are no rational zeros.

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