SOLUTION: Show that the polynomial p(x)= x^4 + 2x^2 + 2x + 2 cannot be expressed as the product of two quadratic polynomials of form x^2 + ax + b and x^2 + cx +d where a,b,c, and d are any i

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Question 32880: Show that the polynomial p(x)= x^4 + 2x^2 + 2x + 2 cannot be expressed as the product of two quadratic polynomials of form x^2 + ax + b and x^2 + cx +d where a,b,c, and d are any integers.
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
F(X)=X^+2X^2+2X+2.....IF THIS HAS TO BE PRODUCT OF 2 QUADRATICS THEN
THEY HAVE TO BE
X^2+AX+1 & X^2-AX+2...OR......
X^2+AX-1 & X^2-AX-2.....AS CONSTANT TERM =2 .ITS INTEGRAL FACTORS ARE
1 &2 OR -1 & -2.
FURTHER COEFFICIENT OF X^3=0 .,SO IF THERE IS AX IN ONE FACTOR,THERE
SHOULD BE -AX IN ANOTHER FACTOR.
THIS LEAVES US....... BY EQUATING COEFFICIENTS OF X AND X^2 TERMS...
(X^2+AX+1)(X^2-AX+2)=X^4-AX^3+2X^2+AX^3-(A^2)(x^2)+2AX+X^2-AX+2
=X^4+X^2(3-A^2)+AX+2=X^4+2X^2+2X+2...SO
3-A^2=2....AND.....A=2.....WHICH ARE NOT CONSISTENT.HENCE THERE IS NO
SOLUTION....SIMILARLY...WE CAN PROVE THIS FOR
(X^2+AX-1)(X^2-AX-2)=X^4-AX^3-2X^2+AX^3-(A^2)(x^2)-2AX-X^2+AX+2
=X^4-X^2(3+A^2)-AX+2=X^4+2X^2+2X+2...SO
3+A^2=2....AND.....A= -2.....WHICH ARE NOT CONSISTENT.HENCE THERE IS
NO SOLUTION....
HENCE F(X) IS NOT REDUCIBLE OR CAN NOT BE FACTORISED INTO 2 QUADRATIC
EXPRESSIONS.