Question

Verify that the numbers given alongside of the Cubic polynomials below are their zeroes.

Also verify the relationship between the zeroes and the Co-efficients in each case.

x^3-4x^2+5x-2;2,1,1.​

Answer 1

Answer:

yes this answer is right you have no answer

Answer 2

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.