Question

What is the HCF of the polynomials x⁶ - 3x⁴ + 3x² - 1 and x³ + 3x² + 3x + 1 ?
1. (x+1)
2. (x+1)²
3. x² + 1
4. (x+1)³

Answer 1

Correct Answer - Option 4 : (x+1)³

Given:

Polynomial f(x) = x6 - 3x4 + 3x2 - 1 

And polynomial g(x) = x3 + 3x2 + 3x + 1

Formula Used:

(x² - y²) = (x + 1)(x - y)

Calculation:

According to the question, we have

f(x) = x6 - 3x4 + 3x2 - 1 

⇒ x⁶ - x⁴ - 2x⁴ + 2x² + x² - 1

⇒ x⁴(x² - 1) - 2x²(x² - 1) +1(x² + 1)

⇒ (x² - 1)(x⁴ - 2x² + 1)

⇒ (x + 1)(x - 1)(x4 - 2x2 + 1)

⇒ (x + 1)(x - 1)(x² - 1)²

⇒ (x + 1) (x - 1) (x + 1)² (x - 1)²

⇒ (x + 1)³(x - 1)³      ----(1)

And,

g(x) = x3 + 3x2 + 3x + 1

⇒ x³ + x² + 2x² + 2x + x + 1

⇒ x²(x + 1) + 2x(x + 1) + 1(x + 1)

⇒ (x + 1)(x² + 2x + 1)

⇒ (x + 1) (x + 1)²

⇒ (x + 1)³      ----(2)

Now,

The HCF of f(x) and g(x) is the common factor of equations (1) and (2)

⇒ (x + 1)³

∴ The HCF of the polynomials x6 - 3x4 + 3x2 - 1 and x3 + 3x2 + 3x + 1 is (x + 1)³.