If the quotient on dividing, 8x⁴ – 2x² + 6x – 7 by 2x + 1 is 4x³ + px² – qx + 3,then find p, q and also the remainder.

Let p(x) = 8x⁴ – 2x² + 6x – 7 be the dividend. Arranging p(x) according to the descending powers of x and write zero in place of missing term.

p(x) = 8x⁴ + 0x³ – 2x² + 6x – 7

Divisor, q(x) = 2x + 1

⇒ To find out Zero of the divisor –

q(x) = 0

2x + 1 = 0

x = .

zero of divisor is .

And, p(x) = 8x⁴ + 0x³ – 2x² + 6x – 7

Put zero for the first entry in the 2^nd row.

∴ Quotient = 8x³ – 4x² + 0x + 6

Hence, when p(x) is divided by (2x + 1) the quotient is 8x³ – 4x² + 0x + 6

6 and remainder is – 10.

Comparing 8x³ – 4x² + 0x + 6 with 4x³ + px² – qx + 3 we get,

p = – 4 and q = 0.