Question

Find the zeros of the following Polynomials by factarisation method and verify the relations between the zeroes and the coefficients of the polynomials ?
$3x {}^{2} + 4x - 4$
​

Answer 1

Step-by-step explanation:

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=

3

2

.

Hence the zeros of the given quadratic polynomial are −2,

3

2

.

Verification of the relationship between the roots and the coefficients:

Sum of the roots =−2+

3

2

=

3

−6+2

=

3

−4

=

coefficientofx

2

−coefficientofx

.

Product of the roots =−2×(

3

2

)

=

3

−4

=

coefficientofx

2

constantterm

.

Therefore, hence verified.

Answer 2

Step-by-step explanation:

Given :-

3X^2+4X-4

To find :-

Find the zeros of the following Polynomials by factarisation method and verify the relations between the zeroes and the coefficients of the polynomials ?

Solution:-

Given quardratic polynomial P(x) = 3X^2+4X-4

Finding the zeroes by factorization method:-

To find the zeroes of P(x) we write P(x) = 0

=>3X^2+4X-4 = 0

=> 3X^2+6X-2X-4 = 0

=> 3X(X+2)-2(X+2) = 0

=> (X+2)(3X-2) = 0

=> (X+2) = 0 or (3X-2) = 0

=>X = -2 or 3X=2

=>X = -2 or X = 2/3

The zeroes of P(x) = -2 and 2/3

ii)Verifying the relationship between the zeroes and the coefficients of P(x) :-

P(x) = 3X^2+4X-4

On Comparing this with the standard quadratic Polynomial ax^2+bx+c

We have ,

a = 3

b = 4

c = -4

and the zeroes = -2 and 2/3

Let α = -2 and β = 2/3

Sum of the zeroes

= α + β

= (-2)+(2/3)

= (-6+2)/3

= -4/3

α + β = -4/3 --------(1)

We know that

Sum of the zeroes = -b/a

=> -4/3------------(2)

From (1)&(2)

α + β = -b/a

Product of the zeroes

= α β

= (-2)×(2/3)

= (-2×2)/3

= -4/3

α β = -4/3----------------(3)

We know that

Product of the zeroes

= c/a

= -4/3-----------(4)

From (3)&(4)

α β = c/a

Answer:-

I) Zeroes of the given quardratic polynomial are -2 and 2/3

i) Verified the relationship between the zeroes and the coefficients of the given quardratic polynomial.

Used Method:-

• Factorization Method

Used formulae:-

• The standard form of a quadratic polynomial is ax^2+bx+c

• If α and β are the zeroes of the quadratic polynomial ax^2+bx+c then

• Sum of the zeroes α + β = -b/a = -(Coefficient of x)/Coefficient of x^2

• Product of the zeroes = α β = c/a = Constant term / Coefficient of x^2