Prove that x4 + 3x3 + 6x2 + 9x + 12 cannot be expressed as a product o

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

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

Answer»

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.

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