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Find the Zeroes of the Following Quadratic Polynomials and Verify the Relationship Between the Zeroes and the Coefficients G(X)=A(X^2+1)-x(A^2+1) - Mathematics

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Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

g(x)=a(x^2+1)-x(a^2+1)

Solution 1

g(x)=a[(x^2+1)-x(a^2+1)]^2=ax^2+a-a^2x-x

=ax^2-[(a^2+1)-x]+0=ax^2-a^2x-x+a

=ax(x-a)-1(x-a)=(x-a)(ax-1)

Zeroes of the polynomials =1/a and a

Sum of zeroes =(-(a^2-1))/a

rArr1/a+a=(a^2+1)/a

rArr(a^2+1)/a=(a^2+1)/a

Product of zeroes =a/a

rArr1/axxa=a/a

rArr1=1

Hence relationship verified

Solution 2

g(x)=a[(x^2+1)-x(a^2+1)]^2=ax^2+a-a^2x-x

=ax^2-[(a^2+1)-x]+0=ax^2-a^2x-x+a

=ax(x-a)-1(x-a)=(x-a)(ax-1)

Zeroes of the polynomials =1/a and a

Sum of zeroes =(-(a^2-1))/a

rArr1/a+a=(a^2+1)/a

rArr(a^2+1)/a=(a^2+1)/a

Product of zeroes =a/a

rArr1/axxa=a/a

rArr1=1

Hence relationship verified

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Find the Zeroes of the Following Quadratic Polynomials and Verify the Relationship Between the Zeroes and the Coefficients G(X)=A(X^2+1)-x(A^2+1) Concept: Relationship Between Zeroes and Coefficients of a Polynomial.