Question

Using remainder and factor theorem, show that (2x + 3) is a factor of the polynomial 2x² + 11x + 12. Hence, factorise it completely. What must be multiplied to the given polynomial so that x² + 3x – 4 is a factor of the resulting polynomial? Also, write the resulting polynomial.

Answer 1

2x + 3 = 0

⇒ 2x = –3

⇒ `x = -3/2`

Substituting value of x in equation 2x² + 11x + 12, we get:

⇒ `2 xx (-3/2)^2 + 11 xx (-3/2) + 12`

⇒ `2 xx 9/4 - 33/2 + 12`

⇒ `9/2 - 33/2 + 12`

⇒ `(9 - 33 + 24)/2`

⇒ `0/2`

⇒ 0

Since, remainder = 0.

∴ 2x + 3 is a factor of the polynomial 2x² + 11x + 12.

Solving polynomial, 2x² + 11x + 12, we get:

⇒ 2x² + 8x + 3x + 12

⇒ 2x(x + 4) + 3(x + 4)

⇒ (2x + 3)(x + 4).

Solving polynomial, x² + 3x – 4, we get:

⇒ x² + 4x – x – 4

⇒ x(x + 4) – 1(x + 4)

⇒ (x – 1)(x + 4).

∴ (x – 1) and (x + 4) are factors of x² + 3x – 4.

∴ On multiplying polynomial, 2x² + 11x + 12 by (x – 1) it will be divisible by x² + 3x – 4.

⇒ (2x² + 11x + 12)(x – 1)

⇒ 2x³ – 2x² + 11x² – 11x + 12x – 12

⇒ 2x³ + 9x² + x – 12.

Hence, the resulting polynomial = 2x³ + 9x² + x – 12.