Question

The HCF and the LCM of two polynomials are 3x + 1 and 30x³ + 7x² - 10x - 3 respectively. If one polynomial is 6x² + 5x + 1, then what is the other polynomial?
1. 15x² + 4x + 3
2. 15x2​ + 4x - 3
3. 15x2​ - 4x + 3
4. ​15x2​ - 4x - 3

Answer 1

Correct Answer - Option 4 : ​15x2​ - 4x - 3

Given:

The HCF of two polynomials = 3x + 1

The LCM of two polynomials = 30x3 + 7x2 - 10x - 3

One polynomial = 6x2 + 5x + 1

Formula used:

The product of two polynomials = Product of their HCF and LCM

Calculation:

Let the two polynomials be f(x) and g(x).

Here, f(x) = 6x2 + 5x + 1

According to the question,

f(x) × g(x) = (3x + 1) × (30x3 + 7x2 - 10x - 3)

⇒ (6x2 + 5x + 1) × g(x) = (3x + 1) × (30x3 + 7x2 - 10x - 3)

⇒ (6x2 + 2x + 3x + 1) × g(x) = (3x + 1) × (30x3 + 15x2 - 8x2 - 4x - 6x - 3)

⇒ [2x(3x + 1) + 1(3x + 1)] × g(x) = (3x + 1)[15x²(2x + 1) - 4x(2x + 1) - 3(2x + 1)] 

⇒ (3x + 1) × (2x + 1) × g(x) = (3x + 1) × (2x + 1) × (15x² - 4x - 3)

⇒ g(x) = (15x2 - 4x - 3)

∴ The other polynomial is 15x2 - 4x - 3.