Consider the following statements :
I. x + 3 is the factor of x^3 + 2x^2 + 3x + 8.
II. x - 2 is the factor of x^3 + 2x^2 + 3x + 8.
Which of the statements given above is/are correct?

Question

Consider the following statements : 
I. x + 3 is the factor of x^3 + 2x^2 + 3x + 8. 
II. x - 2 is the factor of x^3 + 2x^2 + 3x + 8. 
Which of the statements given above is/are correct?

Accepted Answer

To determine whether the given statements about the polynomial $ x^3 + 2x^2 + 3x + 8 $ are correct, we need to verify if $ x + 3 $ and $ x - 2 $ are factors of the polynomial. This can be done using the Factor Theorem, which states that a polynomial $ f(x) $ has a factor $ x - c $ if and only if $ f(c) = 0 $.

**Statement I: $ x + 3 $ is the factor of $ x^3 + 2x^2 + 3x + 8 $.**

To check this, we substitute $ x = -3 $ into the polynomial:

$$ f(-3) = (-3)^3 + 2(-3)^2 + 3(-3) + 8 $$
$$ = -27 + 18 - 9 + 8 $$
$$ = -27 + 18 - 9 + 8 = -10 $$

Since $ f(-3) \neq 0 $, $ x + 3 $ is not a factor of the polynomial. Therefore, Statement I is incorrect.

**Statement II: $ x - 2 $ is the factor of $ x^3 + 2x^2 + 3x + 8 $.**

To check this, we substitute $ x = 2 $ into the polynomial:

$$ f(2) = (2)^3 + 2(2)^2 + 3(2) + 8 $$
$$ = 8 + 8 + 6 + 8 $$
$$ = 30 $$

Since $ f(2) \neq 0 $, $ x - 2 $ is not a factor of the polynomial. Therefore, Statement II is incorrect.

**Conclusion:**

Since neither $ x + 3 $ nor $ x - 2 $ is a factor of the polynomial $ x^3 + 2x^2 + 3x + 8 $, both statements I and II are incorrect. Therefore, the correct answer is Option 4: Neither I nor II. This explanation supports the given correct answer, Option 4, by demonstrating that neither statement satisfies the conditions of the Factor Theorem.