O-Level Differentiation and integration: worked solution

Question

Hence, or otherwise, prove that x³ − 125 = −2x³ + 7x² + 16x − 5 has only one real root.

Bring all terms to one side: x³ - 125 = -2x³ + 7x² + 16x - 5 gives x³ - 125 + 2x³ - 7x² - 16x + 5 = 0, i.e. 3x³ - 7x² - 16x - 120 = 0. Let g(x) = 3x³ - 7x² - 16x - 120. Test x = 5: 3(125) - 7(25) - 16(5) - 120 = 375 - 175 - 80 - 120 = 0, so (x - 5) is a factor. Dividing: 3x³ - 7x² - 16x - 120 = (x - 5)(3x² + 8x + 24). For the quadratic 3x² + 8x + 24, discriminant = 8² - 4(3)(24) = 64 - 288 = -224 < 0, so it has no real roots. Therefore the only real root is x = 5, i.e. the equation has exactly one real root.