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Mathematics

Factors of 4 + 4x - x2 - x3 are :

  1. (2 + x)(2 - x)(1 + x)

  2. (x - 2)(1 + x)(2 + x)

  3. (x + 2)(x - 2)(1 - x)

  4. (2 + x)(x - 1)(2 - x)

Factorisation

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Answer

Substituting x = 2 in 4 + 4x - x2 - x3, we get :

⇒ 4 + 4x - x2 - x3

⇒ 4 + 4(2) - 22 - 23

⇒ 4 + 8 - 4 - 8

⇒ 0.

∴ (x - 2) is the factor of 4 + 4x - x2 - x3.

Dividing -x3 - x2 + 4x + 4 by (x - 2),

x−2)−x2−3x−2x−2)−x3−x2+4x+4‾x−2)2−+x3+−2x2‾x−2x3−2−3x2+4xx−2)x3−2−+3x2+−6x‾x−2)x3−2x2(3)−2x+4x−2)x3−2x2(31)−+2x+−4‾x−2)x3−2x2(31)−2x×\begin{array}{l} \phantom{x - 2)}{-x^2 -3x - 2} \ x - 2\overline{\smash{\big)}-x^3 - x^2 + 4x + 4} \ \phantom{x - 2)}\phantom{2}\underline{\underset{+}{-}x^3 \underset{-}{+}2x^2} \ \phantom{{x - 2}x^3-2}-3x^2 + 4x \ \phantom{{x - 2)}x^3-2}\underline{\underset{+}{-}3x^2 \underset{-}{+} 6x} \ \phantom{{x - 2)}{x^3-2x^{2}(3)}}-2x + 4 \ \phantom{{x - 2)}{x^3-2x^{2}(31)}}\underline{\underset{+}{-}2x \underset{-}{+} 4} \ \phantom{{x - 2)}{x^3-2x^{2}(31)}{-2x}}\times \end{array}

we get quotient = -x2 - 3x - 2.

∴ -x3 - x2 + 4x + 4 = (x - 2)(-x2 - 3x - 2)

= (x - 2)[-x2 - 2x - x - 2]

= (x - 2)[-x(x + 2) - 1(x + 2)]

= (x - 2)(x + 2)(-x - 1)

= -(x - 2)(x + 2)(x + 1)

= (2 - x)(x + 2)(x + 1)

Rearranging the terms we get,

⇒ (2 + x)(2 - x)(1 + x)

Hence, Option 1 is the correct option.

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