What possible numbers of positive roots does this equation have?

Question

x^6+8x^5-3x^4-x^3-7x^2+8x+13=0

Answer 1

The function is x⁶ + 8x⁵ - 3x⁴ - x³ - 7x² + 8x + 13 = 0.

If p/q is a rational zero or root, then p is a factor of constant term (a₀) = 13 and q is a factor of leading coefficient (a_n) = 1.

The possible values of p are   ± 1, and ± 13.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p / q = ± 1, and ± 13.

The possible positive rational roots are 1 and 13.

Substitute the value of x = 1 in the original equation.

(1)⁶ + 8(1)⁵ - 3(1)⁴ - (1)³ - 7(1)² + 8(1) + 13 = 0 ⇒ 19 = 0.

Therefore x = 1 is not a root of the original equation.

Substitute the value of x = 13 in the original equation.

(13)⁶ + 8(13)⁵ - 3(13)⁴ - (13)³ - 7(13)² + 8(13) + 13 = 0

4826809 + 2970344 - 85683 - 2197 - 1183 + 104 + 13 = 0

7797270 - 89063 = 0

7708207 = 0

Therefore x = 13 is not a root of the original equation.

The original equation has no possible positive rational roots.