Question

Simplify each of the following and write as a single polynomial.

(i)

(ii)

(iii)

(iv)

(v)

(v)

Answer 1

(i)

(2x + 1)(3x + 4) + (4x + 3)(3x + 4)

= {(2x + 1) + (4x + 3)}(3x + 4) [Since p(x)r(x) + q(x)r(x) = (p(x) + q(x))r(x)]

= (2x + 4x + 1 + 3)(3x + 4)

= (6x + 4)(3x + 4)

= 18x² + 24x + 12x + 16

= 18x² + 36x + 16

(ii)

(2x + 1)(3x + 4) + (2x − 1)(3x + 4)

= {(2x + 1) + (2x − 1)}(3x + 4) [Since p(x)r(x) + q(x)r(x) = (p(x) + q(x))r(x)]

= (2x + 2x + 1 − 1)(3x + 4)

= 4x(3x + 4)

= 12x² + 16x

(iii)

(2x + 1)(3x + 4) + (1 − 2x)(3x + 4)

= {(2x + 1) + (1 − 2x)}(3x + 4) [Since p(x)r(x) + q(x)r(x) = (p(x) + q(x))r(x)]

= (2x − 2x + 1 + 1)(3x + 4)

= 2(3x + 4)

= 6x + 8

(iv)

(3x + 4)² − (2x − 1) (3x + 4)

= (3x + 4)(3x + 4) − (2x − 1)(3x + 4) [Since p(x)r(x) + q(x)r(x) = (p(x) + q(x))r(x)]

= {(3x + 4) − (2x − 1)}(3x + 4)

= (3x − 2x + 4 + 1)(3x + 4)

= (x + 5) (3x + 4)

= 3x² + 4x + 15x + 20

= 3x² + 19x + 20

(v)

(2x + 1)(3x + 4) + (3x + 4)

= (2x + 1)(3x + 4) + 1(3x + 4)

= {(2x + 1) + 1}(3x + 4) [Since p(x)r(x) + q(x)r(x) = (p(x) + q(x))r(x)]

= (2x + 1 + 1)(3x + 4)

= (2x + 2)(3x + 4)

= 6x² + 8x + 6x + 8

= 6x² + 14x + 8

(vi)

(x² + 3x + 1) (2x + 1) + (x² − 3x − 1) (2x + 1)

= {(x² + 3x + 1) + (x² − 3x − 1)}(2x + 1) [Since p(x)r(x) + q(x)r(x) = (p(x) + q(x))r(x)]

= (x² + 3x + 1 + x² − 3x − 1)(2x + 1)

= 2x²(2x + 1)

= 4x³ + 2x²