Question

find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: (a) 4x^2+5√2x-3​

Answer 1

Answer :

Thus the zeroes of the given polynomial are (1/2√2) and (3/√2)

Given :

The quadratic polynomial is :

• 4x² +5√2x - 3

Task :

• To find the zeroes of the given polynomial.
• To verify the relations between the zeroes and the coefficients of the polynomial.

Solution :

$\sf 4x^{2} +5\sqrt{2}x - 3 \\\\ \sf = 4x^{2} -\sqrt{2}x + 6\sqrt{2}x - 3 \\\\ \sf = \sqrt{2}x (2\sqrt{2}x -1) + 3 (2\sqrt{2}x - 1) \\\\ \sf = (2\sqrt{2}x - 1)(\sqrt{2}x+3)$

Thus the zeroes are :

$\sf 2\sqrt{2} x-1=0 \: \: and \: \: \sqrt{2}x +3=0 \\\\ \sf \implies x = \dfrac{1}{2\sqrt{2}} \: \: and \: \implies x =- \dfrac{3}{\sqrt{2}}$

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Verification of the relations of the zeroes and coefficients :

$\sf \dashrightarrow Sum \: of \: the \: zeroes =-\dfrac{Coefficient of x }{Coefficient of x^{2}} \\\\ \sf \implies \dfrac{1}{2\sqrt{2}} + (-\dfrac{3}{\sqrt{2}})=- \dfrac{5\sqrt{2}}{4} \\\\ \sf\implies \dfrac{1 - 6}{2\sqrt{2}} = -\dfrac{5}{2\sqrt{2}} \\\\ \sf \implies -\dfrac{5}{2\sqrt{2}} = -\dfrac{5}{2\sqrt{2}}$

$\sf \dashrightarrow Product \: of \: the \: zeroes = \dfrac{Constant \: term}{Coefficient \: of \: x^{2}} \\\\ \sf \implies \dfrac{1}{2\sqrt{2}}\times (-\dfrac{3}{\sqrt{2}}) = \dfrac{-3}{4} \\\\ \sf \implies -\dfrac{3}{4} =-\dfrac{3}{4}$

Hence verified