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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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