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ICSE Class 10 Mathematics Question 8 of 28

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8
Question

Question 57

Pamela factorized the following polynomial :

2x3 + 3x2 - 3x - 2

She found the result as (x + 2)(x - 1)(x - 2). Using remainder and factor theorem, verify whether her result is correct. If incorrect, give the correct result.

Answer

Dividing the polynomial f(x) = 2x3 + 3x2 - 3x - 2 by (x + 2), we get :

⇒ x + 2 = 0

⇒ x = -2

f(-2) = 2(-2)3 + 3(-2)2 - 3(-2) - 2

= 2(-8) + 3(4) + 6 - 2

= -16 + 12 + 6 - 2

= -18 + 18

= 0.

Dividing the polynomial f(x) = 2x3 + 3x2 - 3x - 2 by (x - 1), we get :

⇒ x - 1 = 0

⇒ x = 1

f(1) = 2(1)3 + 3(1)2 - 3(1) - 2

= 2(1) + 3(1) - 3 - 2

= 2 + 3 - 3 - 2

= 0.

Dividing the polynomial f(x) = 2x3 + 3x2 - 3x - 2 by (x - 2), we get :

⇒ x - 2 = 0

⇒ x = 2

f(2) = 2(2)3 + 3(2)2 - 3(2) - 2

= 2(8) + 3(4) - 6 - 2

= 16 + 12 - 6 - 2

= 20.

Since, f(2) ≠ 0

∴ x - 2 does not divide the polynomial 2x3 + 3x2 - 3x - 2.

Dividing the polynomial f(x) by (x + 2)(x - 1) or by (x2 + x - 2), we get :

x2+x−2)2x+1x2+x−2)2x3+3x2−3x−2‾x2+x−2))+−2x3+−2x2−+4x‾x2+x−22x3+20x3x2+x−2x2+x−2)2x3+x3+−x2+−x−+2‾x2+x−2)x3−2x2(31)x×\begin{array}{l} \phantom{x^2 + x - 2)}{\quad 2x + 1} \\ x^2 + x - 2\overline{\smash{\big)}\quad 2x^3 + 3x^2 - 3x - 2} \\ \phantom{x^2 + x - 2)}\phantom{)}\underline{\underset{-}{+}2x^3 \underset{-}{+}2x^2 \underset{+}{-}4x} \\ \phantom{{x^2 + x - 2}2x^3 + 20x^3}x^2 + x - 2 \\ \phantom{{x^2 + x - 2)}2x^3 + x^3}\underline{\underset{-}{+}x^2 \underset{-}{+} x \underset{+}{-} 2} \\ \phantom{{x^2 + x - 2)}{x^3-2x^{2}(31)}{x}}\times \end{array}

∴ 2x3 + 3x2 - 3x - 2 = (x2 + x - 2)(2x + 1)

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

Hence, 2x3 + 3x2 - 3x - 2 = (x + 2)(x - 1)(2x + 1).