Using remainder and factor theorem, factorize completely, the given polynomial :
2x3 - 9x2 + 7x + 6.
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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).