Question

Find the zeroes of the following polynomials by factorisation method and verify the relation between the zeroes and the coefficients of the polynomials:$$3x^2 + 4x- 4$$.

Solution

Let $$f(x)=3x^{ 2 }+4x-4$$.Comparing it with the standard quadratic polynomial $$ax^2+bx+c$$, we get,$$a=3$$, $$b=4$$, $$c=-4$$.Now, $$3x^{ 2 }+4x-4$$$$=3x^{ 2 }+6x-2x-4$$$$=3x(x+2)-2(x+2)$$$$=(x+2)(3x-2)$$.The zeros of $$f(x)$$ are given by $$f(x)=0$$.$$=>(x+2)(3x-2)=0$$$$=>x+2=0, 3x-2=0$$$$=>x=-2, x=\dfrac { 2}{ 3}$$.Hence the zeros of the given quadratic polynomial are $$-2$$, $$\dfrac { 2}{ 3}$$.Verification of the relationship between the roots and the coefficients:Sum of the roots $$=-2+\dfrac { 2}{ 3}$$                             $$=\dfrac { -6+2}{ 3}$$                             $$=\dfrac { -4}{ 3}$$                             $$=\dfrac { -coefficient\quad of\quad x }{ coefficient\quad of\quad { x }^{ 2 } }$$.Product of the roots $$= -2\times (\dfrac { 2 }{ 3 })$$                                   $$=\dfrac { -4 }{ 3 }$$                                   $$=\dfrac { constant\quad term }{ coefficient\quad of\quad { x }^{ 2 } }$$.Therefore, hence verified.Mathematics

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