Question

Verify that the numbers given along side of the cubic polynomials below are their zeros.Also verify the relationship between thw zeros and coefficients in each case:g(x)=x^3-4x^2+5x-2;,2,1,1

Answer 1

p(x)=x

3

−4x

2

+5x−2 .... (1)

Zeroes for this polynomial are 2,1,1

Substitute x=2 in equation (1)

p(2)=2

3

−4×2

2

+5×2−2

=8−16+10−2=0

Substitute x=1 in equation (1)

p(1)=x

3

−4x

2

+5x−2

=1

3

−4(1)

2

+5(1)−2

=1−4+5−2=0

Therefore, 2,1,1 are the zeroes of the given polynomial.

Comparing the given polynomial with ax

3

+bx

2

+cx+d we obtain,

a=1,b=−4,c=5,d=−2

Let us assume α=2, β=1, γ=1

Sum of the roots = α+β+γ=2+1+1=4=−

1

−4

a

−b

Multiplication of two zeroes taking two at a time=αβ+βγ+αγ=(2)(1)+(1)(1)+(2)(1)=5=

1

5

=

a

c

Product of the roots = αβγ=2×1×1=2=−

1

−2

=

a

d

Therefore, the relationship between the zeroes and coefficient are verified.