Algebra

Advanced Factorization

Factorization of Quadratics

         

If xy5=0x-y-5=0, then the polynomial 3x2+xyy225xy+503x^2+xy-y^2-25x-y+50 can be simplified as ay(y+b)ay(y+b). What is a+ba+b?

Given that x=a+b+c,y=ab+c,z=a+bc,a2+b2+c2=20,\begin{aligned} & x = -a+b+c, \\ & y = a-b+c, \\ & z = a+b-c ,\\ & a^2+b^2+c^2 =20, \end{aligned} find the value of x2+y2+z2+xy+yz+zxx^2+y^2+z^2+xy+yz+zx.

Suppose that aa is an integer such that x2+2xy80y2+ax+18y+81x^2+2xy-80y^2+ax+18y+81 can be factorized as the product of linear expressions in xx and yy. What is the value of aa?

a,ba, b and cc are integers such that (xa)(x20)+1=(x+b)(x+c). (x-a)(x-20) + 1 = (x+b)(x+c) . What is the sum of all possible values of a a ?

Suppose aa, bb and cc are real numbers such that a+b+c=5.a+b+c=5. If x=a2b+3c,y=b2c+3a,z=c2a+3b,x=a-2b+3c, y=b-2c+3a, z=c-2a+3b, what is the value of (x2+2xy+4)+(y2+2yz+4)+(z2+2zx+4)?(x^2+2xy+4)+(y^2+2yz+4)+(z^2+2zx+4)?

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