Given Px 2x4 15x3 39x2 41x 15 a List all the possible r

Given P(x) = 2x^4 - 15x^3 + 39x^2 - 41x + 15, (a) List, all the possible rational zeros of P(x).

Solution

P(x) = 2x⁴ - 15x³ + 39x² - 41x + 15

By observation, P(1) =0 => (x - 1) is a factor of P(x)

Hence,

P(x) = 2x⁴ - 15x³ + 39x² - 41x + 15

= 2x⁴ - 2x³ - 13x³ + 13x² + 26x² - 26x - 15x +15

= 2x³(x-1) - 13x²(x-1) +26x(x-1) -15(x-1)

= (2x³ - 13x² + 26x - 15)(x-1)

=(x-1) { 2x³ - 2x² - 11x² + 11x + 15x -15 }

=(x-1) { 2x²(x-1) - 11x(x-1) + 15(x-1)}

=(x-1)²(2x² -11x + 15)

=(x-1)²(2x² - 6x - 5x + 15)

=(x-1)² { 2x (x-3) - 5(x-3) }

= (x-1)²(x-3)(2x-5)

Hence P(x) = 0 has 4 rational zeros and they are 1, 1, 3, 5/2.