Question

Prove that x⁴ + 3x³ + 6x² + 9x + 12 cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.

Answer 1

Let x⁴ + 3x³ + 6x² + 9x + 12

= (x² + Ax + B) (x² + Cx + D)

= x⁴ + Cx³ + Dx² + Ax³ + ACx² + ADx + Bx² + BCx + BD

= x⁴ + (A + C)x³ + (D + AC + B) x² + (AD + BC)x + BD

Now by comparing coefficient

A + C = 3

B + D + AC = 6

AD + BC = 9

BD = 12

Case - I : B = 1, D = 12

∴ A + C = 3

12A + C = 9 have no integer solution.

Case - II : B = - 1, D = - 12

C + 12 A = - 9

C + A = 3 have no integer solution.

Case - III : B = 2, D = 6

2C + 6A = 9

C + A = 3 have no integer solution.

Case - IV : B = - 2, D = - 6

2C + 6A = - 9

A + C = 3 have no integer solution.

So, x⁴ + 3x³ + 6x² + 9x + 12 cannot be expressed as a product of two polynomial of degree 2 with integer coefficient.