Using the Rational Root Theorem, what are all the rational roots of the polynomial f(x) = 20x4 + x3 + 8x2 + x – 12?

We have,

f(x) = 20x⁴ + x³ + 8x² + x – 12

The Rational Root Theorem states that the all possible roots of a polynomial are in the form of a rational number i.e in the form of p/q.

Where, 

 p = a factor of constant term 

and q =  the factor of coefficient of leading term

Here, constant p = -12

Now, factors of -12 are ±1,±2,±3,±4,±6,±12

And, leading term, q = 20

Now, factors of 20 are ±1,±2,±4,±5,±10,±20

So, all possible factors of polynomial are,

±1,±2,±3,±4,±6,±12,±1/2,±3/2,±1/4,±3/4,±1/10,±1/5,±3/5,±3/10,±2/5,±6/5,±1/20,±3/20,±4/5,±12/5

Rational roots of polynomial that satisfy the given polynomial:

f(-4/5) = 20(-4/5)⁴ + (-4/5)³ + 8(-4/5)² + (-4/5) - 12

= 256/625 × 20 - 64/125 + 8 × 16/25 - 4/5 - 12

= 1024/125 - 64/125 + 128/25 -4/5 - 12

= 960/125 + 128/25 - 4/5 - 12

= 12 - 12 

= 0

Now, for x = 3/4

f(3 /4) = 20(3 /4)⁴ + (3 /4)³ + 8(3 /4)² + (3 /4) -12

= 20 × 81/256 + 27/64 + 8 × 9/16 + 3 /4 -12

= 405/64 + 27/64 + 9/2 + 3 /4 - 12

= (405 + 27 + 288 + 48 - 768)/64

= (768 - 768)/64

= 0

Hence, factors of f(x) = 20x4 + x3 + 8x2 + x – 12 are -4/5, 3 /4