Find the value of p and q so that x4 + px3 + 2x2 - 3x + q is divisi

1.	Find the value of p and q so that x4 + px3 + 2x2 - 3x + q is divisible by (x2-1).
Answer» Let, f (x) = x⁴ + px³ + 2x² - 3x + q be the given polynomial. And, let g (x) = (x² – 1) = (x – 1) (x + 1) Clearly, (x – 1) and (x + 1) are factors of g (x) Given, g (x) is a factor of f (x) (x – 1) and (x + 1) are factors of f (x) From factor theorem If (x – 1) and (x + 1) are factors of f (x) then f (1) = 0 and f (-1) = 0 respectively. f (1) = 0 (1)⁴ + p (1)³ + 2 (1)²– 3 (1) + q = 0 1 + p + 2 – 3 + q = 0 p + q = 0 (i) Similarly, f (-1) = 0 (-1)⁴ + p (-1)³ + 2 (-1)² - 3 (-1) + q = 0 1 – p + 2 + 3 + q = 0 q – p + 6 = 0 (ii) Adding (i) and (ii), we get p + q + q – p + 6 = 0 2q + 6 = 0 2q = - 6 q = -3 Putting value of q in (i), we get p – 3 = 0 p = 3 Hence, x² – 1 is divisible by f (x) when p = 3 and q = - 3.

Find the value of p and q so that x4 + px3 + 2x2 - 3x + q is divisible by (x2-1).

Answer»

Let, f (x) = x⁴ + px³ + 2x² - 3x + q be the given polynomial.

And, let g (x) = (x² – 1) = (x – 1) (x + 1)

Clearly,

(x – 1) and (x + 1) are factors of g (x)

Given,

g (x) is a factor of f (x)

(x – 1) and (x + 1) are factors of f (x)

From factor theorem

If (x – 1) and (x + 1) are factors of f (x) then f (1) = 0 and f (-1) = 0 respectively.

f (1) = 0

(1)⁴ + p (1)³ + 2 (1)²– 3 (1) + q = 0

1 + p + 2 – 3 + q = 0

p + q = 0 (i)

Similarly,

f (-1) = 0

(-1)⁴ + p (-1)³ + 2 (-1)² - 3 (-1) + q = 0

1 – p + 2 + 3 + q = 0

q – p + 6 = 0 (ii)

Adding (i) and (ii), we get

p + q + q – p + 6 = 0

2q + 6 = 0

2q = - 6

q = -3

Putting value of q in (i), we get

p – 3 = 0

p = 3

Hence,

x² – 1 is divisible by f (x) when p = 3 and q = - 3.