What are all the rational roots of the polynomial f(x) = 20x⁴ + x³ + 8x² + x - 12?

Solution:

f(x) = 20x⁴ + x³ + 8x² + x - 12 (Given)

According to the Rational Root Theorem, all the possible roots of a polynomial are in the form of a rational number.

x = p/q

Where p is a factor of a constant term

q is the factor of coefficient of leading term

From the polynomial given

-12 is the constant term and 20 is the coefficient of leading term

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

The possible factors of 20 are ±1, ±2, ±4, ±5, ±10, ±20

So the rational roots = (±1, ±2, ±3, ±4, ±6, ±12)/ (±1, ±2, ±4, ±5, ±10, ±20)

Solve for x when the given function is equal to zero

20x⁴ + x³ + 8x² + x - 12 = 0

(4x - 3)(5x + 4)(x² + 1) = 0

So we get

4x - 3 = 0 or 5x + 4 = 0 or x² + 1 = 0

x = 3/4, x = -4/5, x = i, x = -i

Hence, the rational roots are x = 3/4, x = -4/5.

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What are all the rational roots of the polynomial f(x) = 20x⁴ + x³ + 8x² + x - 12?

Summary:

All the rational roots of the polynomial f(x) = 20x⁴ + x³ + 8x² + x - 12 are 3/4 and -4/5.