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Mathematics

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.

Factorisation

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Answer

2x + 3 = 0

⇒ 2x = -3

⇒ x = −32-\dfrac{3}{2}

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

⇒2×(−32)2+11×(−32)+12⇒2×94−332+12⇒92−332+12⇒9−33+242⇒02⇒0.\Rightarrow 2 \times \Big(-\dfrac{3}{2}\Big)^2 + 11 \times \Big(-\dfrac{3}{2}\Big) + 12 \\[1em] \Rightarrow 2 \times \dfrac{9}{4} - \dfrac{33}{2} + 12 \\[1em] \Rightarrow \dfrac{9}{2} - \dfrac{33}{2} + 12 \\[1em] \Rightarrow \dfrac{9 - 33 + 24}{2} \\[1em] \Rightarrow \dfrac{0}{2} \\[1em] \Rightarrow 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.

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