While factorizing a given polynomial, using remainder and factor theorem, a student finds that x + 3 is a factor of 2x³ - x² - 5x - 2.

(a) Is the student's, solution correct stating that (x + 3) is a factor of the given polynomial?

(b) Give a valid reason for your answer.

(c) Factorize the given polynomial completely.

Given polynomial: 2x³ - x² - 5x - 2.

By factor theorem,

If x - a is the factor of polynomial f(x), then remainder f(a) = 0.

If (x + 3) is a factor, then by Factor Theorem :

f(−3) = 0.

Substituting x = -3 in polynomial we get, remainder = 0 :

⇒ 2(-3)³ - (-3)² - 5(-3) - 2

⇒ 2(-27) - 9 + 15 - 2

⇒ -54 - 9 + 15 - 2

⇒ -50.

Since f(-3) ≠ 0, Remainder ≠ 0.

Hence, the student's solution is incorrect.

Factorizing,

Substituting x = 2 in polynomial we get,

f(2) = 2(2)³ - (2)² - 5(2) - 2

= 16 - 4 - 10 - 2

= 16 - 16

= 0.

Since f(2) = 0, (x - 2) is a factor 2x³ - x² - 5x - 2.

On dividing, 2x³ - x² - 5x - 2 by x - 2,

$\begin{array}{l} \phantom{x - 2)}{2x^2 + 3x + 1} \ x - 2\overline{\smash{\big)}2x^3 - x^2 - 5x - 2} \ \phantom{x - 1}\underline{\underset{-}{}2x^3 \underset{+}{-} 4x^2} \ \phantom{{x - 1}3xm.^30}3x^2 - 5x \ \phantom{{x - 1}3.3+}\underline{\underset{-}{}3x^2 \underset{+}{-} 6x} \ \phantom{{x - 1}{3x^3+3x^2+}{+2}}x - 2 \ \phantom{{x - 1}{3x^3+3x^2+}\enspace \space}\underline{\underset{-}{}x \underset{+}{-} 2} \ \phantom{{x - 1}{3x^3+3x^2+}{+2-}}\times \end{array}$

2x³ - x² - 5x - 2 = (x - 2)(2x² + 3x + 1)

= (x - 2)(2x² + 2x + x + 1)

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

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

Hence, 2x³ - x² - 5x - 2 = (x - 2)(x + 1)(2x + 1).