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Consider the following statements: 1. x + 3 is a factor of x3 + 2x2 + 3x + 8 2. x – 2 is a factor of x3 + 2x2 + 3x + 8 Which of the statements given above is/are correct? 

Consider the following statements: 1. x + 3 is a factor of x3 + 2x2 + 3x + 8 2. x – 2 is a factor of x3 + 2x2 + 3x + 8 Which of the statements given above is/are correct?  Correct Answer Neither 1 nor 2

Concept:

Factor theorem:

If f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number, then,

(x - a) is a factor of f(x), if f(a) = 0.

  •  If f(-a) = 0, then (x + c) is a factor of the polynomial f(x).
  •  If f(d/c) = 0, then (cx - d) is a factor of the polynomial f(x).
  •  If f(-d/c) = 0, then (cx + d) is a factor of the polynomial f(x).
  •  If f(a) = 0 and f(b) = 0, then (x - a) and (x - b) are factors of the polynomial p(x).

 

Calculation:

Considering the given equation

f(x) = x3 + 2x2 + 3x + 8    -----(1)

for x + 3 = 0 ⇒ x = - 3

f(-3) = (-3)3 + 2 × (-3)2 + 3 × (-3) + 8 = - 27 + 18 – 9 + 8 

⇒  f(-3) = -10 ≠ 0

⇒  (x + 3) is not a factor of x3 + 2x2 + 3x + 8

for x – 2 = 0 ⇒ x = 2

f(2) = (2)3 + 2 × (2)2 + 3 × 2 + 8 = 8 + 8 + 6 + 8 

⇒  f(2) = 30 ≠ 0

⇒  (x – 2) is not a factor of x3 + 2x2 + 3x + 8

∴ Neither 1 nor 2 is correct

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