Question

Discuss the maximum possible number of positive and negative roots of the polynomial equations x² – 5x + 6 and x² – 5x + 16. Also, draw a rough sketch of the graphs

Answer 1

y = x² – 5x + 6

x = 1, y = 1 – 5 + 6 = 2

x = 2, y = 4 – 10 + 6 = 0

x = 0, y = 6

x = 3, y = 9 – 15 + 6 = 0

x = – 1, y = 1 + 5 + 6 = 12

x = 4, y = 16 – 20 + 6 = 2

(1, 2), (0, 6), (– 1, 12)

P(x) = (x² + 5x + 6)(x² – 5x + 16)

= x⁴ – 5x³ + 16x² – 5x³ + 25x² – 80x + 6x² – 30x + 96

= 0

= x⁴ – 10x³ + 47x² – 110x + 96 = 0

It has two sign changes

∴ It has two positive real roots

P(– x) = x⁴ + 10x³ + 47x² + 110x + 96

It has no sign change, no negative real roots

y = x² – 5x + 16

x	0	1	–1	2	4
y	16	12	23	10	12