Solved 2024 Specimen Paper ICSE Class 10 Mathematics — Question 18

Substituting x = 2, in the given polynomial, we get :

⇒ 2(2)³ - 9(2)² + 7(2) + 6

⇒ 2 × 8 - 9 × 4 + 14 + 6

⇒ 16 - 36 + 20

⇒ -20 + 20

⇒ 0.

∴ x - 2 is the factor of the given polynomial.

On dividing 2x³ - 9x² + 7x + 6 by x - 2, we get :

$\begin{array}{l} \phantom{x - 2)}{\quad 2x^2 - 5x - 3} \\ x - 2\overline{\smash{\big)}\quad 2x^3 - 9x^2 + 7x + 6} \\ \phantom{x - 2)}\phantom{)}\underline{\underset{-}{+}2x^3 \underset{+}{-}4x^2} \\ \phantom{{x - 2}2x^3-4}-5x^2 + 7x \\ \phantom{{x - 2)}2x^3-4}\underline{\underset{+}{-}5x^2 \underset{-}{+} 10x} \\ \phantom{{x - 2)}31x^3-2+1}-3x + 6 \\ \phantom{{x - 2)}31x^3-2+11}\underline{\underset{+}{-}3x \underset{-}{+} 6} \\ \phantom{{x - 2)}31x^3-2+11+1}\times \end{array}$

∴ 2x³ - 9x² + 7x + 6 = (x - 2)(2x² - 5x - 3)

= (x - 2)[2x² - 6x + x - 3]

= (x - 2)[2x(x - 3) + 1(x - 3)]

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

Hence, 2x³ - 9x² + 7x + 6 = (x - 2)(2x + 1)(x - 3).