Question

Find the quotient and remainder when each of the polynomials below is divided by 2x + 1.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Answer 1

(i)

6x² + 7x + 3

= (6x² + 7x + 2) + 1

= (6x² + 4x + 3x + 2) + 1

= 2x(3x + 2) + 1(3x + 2) + 1

= (3x + 2)(2x + 1) + 1

Now,

∴ Quotient is 3x + 2 and remainder is 1.

(ii)

6x² + 7x − 1

= (6x² + 7x + 2) − 3

= (6x² + 4x + 3x + 2) − 3

= 2x(3x + 2) + 1(3x + 2) − 3

= (3x + 2)(2x + 1) − 3

Now,

∴ Quotient is 3x + 2 and remainder is − 3.

(iii)

6x² + 9x + 3

= 6x² + 6x + 3x + 3

= 6x(x + 1) + 3(x + 1)

= (6x + 3)(x + 1)

= 3(2x + 1)(x + 1)

Now,

∴ Quotient is 3x + 3 and remainder is 0.

(iv)

6x² + 9x + 4

= 6x² + 9x + 3 + 1

= (6x² + 6x + 3x + 3) + 1

= 6x(x + 1) + 3(x + 1) + 1

= (6x + 3)(x + 1) + 1

= 3(2x + 1)(x + 1) + 1

Now,

∴ Quotient is 3x + 3 and remainder is 1.

(v)

6x² + 9x + 2

= 6x² + 9x + 3 −1

= (6x² + 6x + 3x + 3) − 1

= 6x(x + 1) + 3(x + 1) −1

= (6x + 3)(x + 1) − 1

= 3(2x + 1)(x + 1) − 1

Now,

∴ Quotient is 3x + 3 and remainder is −1.

(vi)

Now,

∴ Quotient is and remainder is 0.