If α and β are the zeroes of the quadratic polynomial f(x) = 2x² - 5x + 7 . find a quadratic polynomial whose zeroes are 2α+3β and 3α+ 2β
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a+b=5/2______(1)
ab=7/2________(2)
now zeroes of new polynomial
2a+3b+3a+2b=5a+5b=5(a+b)=5×5/2 (using 1)
=25/2
now
(2a+3b)(3a+2b)=6a^2+4ab+9ab+6b^2
=6 (a^2+b^2)+13ab
=6 (a+b )^2-2ab+13an
=6×25/4+11×7/2
=75/2+77/2
=152/2=76
so new polynomial is
x^2-25/2+76
or 2x^2-25+152
ab=7/2________(2)
now zeroes of new polynomial
2a+3b+3a+2b=5a+5b=5(a+b)=5×5/2 (using 1)
=25/2
now
(2a+3b)(3a+2b)=6a^2+4ab+9ab+6b^2
=6 (a^2+b^2)+13ab
=6 (a+b )^2-2ab+13an
=6×25/4+11×7/2
=75/2+77/2
=152/2=76
so new polynomial is
x^2-25/2+76
or 2x^2-25+152
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