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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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