Question

Divide the first polynomial by the second polynomial and find the remainder using factor theorem .

(i) $\left(x^2-7x+9\right);(x+1)$ (ii) $\left(2x^3-2x^2+ax-a\right);(x-a)$ (iii) $\left(54m^3+18m^2-27m+5\right);(m-3)$

Answer 1

(i)
By synthetic division:

Dividend = $x^2-7x+9$

Divisor = x + 1

Opposite of 1 = −1



The coefficient form of the quotient is (1, −8).

∴ Quotient = x − 8

Remainder = 17

By remainder theorem:

Let $p\left(x\right)=x^2-7x+9$.

Divisor = x + 1

By remainder theorem,

Remainder = p(−1) = ${\left(-1\right)}^2-7\times \left(-1\right)+9=1+7+9=19$

(ii)
By synthetic division:

Dividend = $2x^3-2x^2+ax-a$

Divisor = x − a

Opposite of −a = a



The coefficient form of the quotient is (2, 2a − 2, 2a²​ − a).

∴ Quotient = 2x² + (2a − 2)x + 2a²​ − a

Remainder = 2a³​ − a² − a

By remainder theorem:

Let $p\left(x\right)=2x^3-2x^2+ax-a$.

Divisor = x − a

By remainder theorem,

Remainder = p(a) = $2\times a^3-2\times a^2+a\times a-a=2a^3-2a^2+a^2-a=2a^3-a^2-a$

(iii)
By synthetic division:

Dividend = $54m^3+18m^2-27m+5$

Divisor = m − 3

Opposite of −3 = 3



The coefficient form of the quotient is (54, 180, 513).

∴ Quotient = 54x² + 180x + 513

Remainder = 1544

By remainder theorem:

Let $p\left(m\right)=54m^3+18m^2-27m+5$.

Divisor = m − 3

By remainder theorem,

Remainder = p(3) = $54\times {\left(3\right)}^3+18\times {\left(3\right)}^2-27\times 3+5=54\times 27+18\times 9-27\times 3+5=1458+162-81+5=1544$