Question

Find the value of p and q if (a²-4) is a factor of pa⁴+2a³-3a²+qa-4

Answer 1

Solution :

Given,

a² - 4 = 0

a² = 4

a = √4

a = ±2

pa⁴ + 2a³ - 3a² + qa - 4

I) Substitute 2 in the place of a.

p(2)⁴ + 2(2)³ - 3(2)² + q(2) - 4 = 0

p(16) + 2(8) - 3(4) + 2q - 4 = 0

16p + 16 - 12 + 2q - 4 = 0

16p + 2q + 16 - 16 = 0

16p + 2q = 0 ----(1)

ii) Substitute -2 in the place of a

p(-2)⁴ + (-2)³ - 3(-2)² + q(-2) - 4 = 0

16p + (-8) - 3(4) - 2q - 4 = 0

16p - 8 - 12 - 2q - 4 = 0

16p - 24 - 2q = 0

16p - 2q = 24 ---------(2)

Solve 1 & 2

16p + 2q = 0

2p - 2q = 24

-----------------

18p = 24

p = 24/18

p = 4/3

Substitute p in eq - (1)

16p + 2q = 0

16(4/3) + 2q = 0

64/3 + 2q = 0

2q = - 64/3

q = -64/3 × 1/2

q = -64/6

q = -32/3

Therefore, the values of p and q are 2/3 and -32/3 respectively.