Divide the first polynomial by the second polynomial and find the rema

1.	Divide the first polynomial by the second polynomial and find the remainder using remainder theorem. i. (x2 – 1x + 9); (x + 1) ii. (2x3 – 2x2 + ax – a); (x – a) iii. (54m3 + 18m2 – 27m + 5); (m – 3)
Answer» i. p(x) = x² – 7x + 9 Divisor = x + 1 ∴ take x = – 1 ∴ By remainder theorem, ∴ Remainder =p(-1) p(x) = x² – 7x + 9 ∴ p(-1) = (- 1)² – 7(- 1) + 9 = 1 + 7 + 9 ∴ Remainder = 17 ii. p(x) = 2x³ – 2x² + ax – a Divisor = x – a ∴ take x = a By remainder theorem, Remainder = p(a) p(x) = 2x³ – 2x² + ax – a ∴ p(a) = 2a³ – 2a² + a(a) – a = 2a³ – 2a² + a² – a ∴ Remainder = 2a³ – a² – a iii. p(m) = 54m³ + 18m² – 27m + 5 Divisor = m – 3 ∴ take m = 3 ∴ By remainder theorem, Remainder = p(3) p(m) = 54m³ + 18m² – 27m + 5 ∴ p(3) = 54(3)³ +18(3)² – 27(3) + 5 = 54(27) + 18(9) – 81 + 5 = 1458 + 162 – 81 + 5 ∴ Remainder = 1544

Divide the first polynomial by the second polynomial and find the remainder using remainder theorem. i. (x2 – 1x + 9); (x + 1) ii. (2x3 – 2x2 + ax – a); (x – a) iii. (54m3 + 18m2 – 27m + 5); (m – 3)

Answer»

i. p(x) = x² – 7x + 9

Divisor = x + 1

∴ take x = – 1

∴ By remainder theorem,

∴ Remainder =p(-1)

p(x) = x² – 7x + 9

∴ p(-1) = (- 1)² – 7(- 1) + 9

= 1 + 7 + 9

∴ Remainder = 17

ii. p(x) = 2x³ – 2x² + ax – a

Divisor = x – a

∴ take x = a

By remainder theorem,

Remainder = p(a)

p(x) = 2x³ – 2x² + ax – a

∴ p(a) = 2a³ – 2a² + a(a) – a

= 2a³ – 2a² + a² – a

∴ Remainder = 2a³ – a² – a

iii. p(m) = 54m³ + 18m² – 27m + 5

Divisor = m – 3

∴ take m = 3

∴ By remainder theorem,

Remainder = p(3)

p(m) = 54m³ + 18m² – 27m + 5

∴ p(3) = 54(3)³ +18(3)² – 27(3) + 5

= 54(27) + 18(9) – 81 + 5

= 1458 + 162 – 81 + 5

∴ Remainder = 1544