Prove that x4 + 3x3 + 6x2 + 9x + 12 cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
Lost your password? Please enter your email address. You will receive a link and will create a new password via email.
Please briefly explain why you feel this question should be reported.
Please briefly explain why you feel this answer should be reported.
Please briefly explain why you feel this user should be reported.
Prove that x4 + 3x3 + 6x2 + 9x + 12 cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
Prove that x4 + 3x3 + 6x2 + 9x + 12 cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
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.