Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) \(2x^3 + x^2 - 5x + 2 ; {{1}\over{2}},1,-2\)
(ii) \(x^3 - 4x^2 + 5x - 2 ; 2,1,1\)

(i)The general form of cubic equation is \(ax^3 + bx^2 + cx + d\).

Here, \(a = 2,b = 1,c = -5,d = 2\)

On substituting the value of zeroes in the given equation, p(x)= \(2x^3 + x^2 - 5x + 2\)

p\(({{1}\over{2}})\)= \(2({{1}\over{2}})^3 + ({{1}\over{2}})^2 + -5({{1}\over{2}}) + 2\)
=>\({{1}\over{4}} + {{1}\over{4}} - {{5}\over{2}} + 2 = 0 \)
p\((1)\)= \(2(1)^3 + (1)^2 + -5(1) + 2\)
=>\(2 + 1 - 5 + 2 = 0 \)
p\((-2)\)= \(2(-2)^3 + (-2)^2 + -5(-2) + 2\)
=>\(-16 + 4 + 10 + 2 = 0 \)

Hence, these zeroes satisfies the the given equation.
Now, we'll check whether the zeroes satisfies the following equations:

Let the three zeroes be p,q and r
\(p + q + r\) = \({{-b}\over{a}}\)............(i)
\(pq + qr + rp\) = \({{c}\over{a}}\)..........(ii)
\(pqr\) = \({{-d}\over{a}}\).................(iii)
On checking equation (i):=> \({{1}\over{2}} + 1 + (-2)\)
=>\({{3}\over{2}} - 2 \)
=> \({{-1}\over{2}} = {{-b}\over{a}}\)

On checking equation (ii):=> \({{1}\over{2}} × 1 + 1 × (-2) + (-2) × {{1}\over{2}}\)
=>\({{1}\over{2}} - 2 - 1\)
=> \({{-5}\over{2}} = {{c}\over{a}}\)
On checking equation (iii):=> \({{1}\over{2}} × 1 × (-2)\)
=>\({{-2}\over{2}}\)
=> \( -1 = {{-d}\over{a}}\)

Hence it is verified the numbers given alongside of the cubic polynomials are their zeroes.

(ii) The general form of cubic equation is \(ax^3 + bx^2 + cx + d\).

Here, \(a = 1,b = -4,c = 5,d = -2\)

On substituting the value of zeroes in the given equation, p(x)= \(x^3 - 4x^2 + 5x - 2 \)

p\((2)\)= \(2^3 - 4(2)^2 + 5(2) - 2 \)
=>\(8 - 16 + 10 - 2 = 0 \)
p\((1)\)= \(1^3 - 4(1)^2 + 5(1) - 2\)
=>\(1 - 4 + 5 - 2 = 0\)

Hence, these zeroes satisfies the the given equation.

Now, we'll check whether the zeroes satisfies the following equations:

Let the three zeroes be p,q and r
\(p + q + r\) = \({{-b}\over{a}}\)............(i)
\(pq + qr + rp\) = \({{c}\over{a}}\)..........(ii)
\(pqr\) = \({{-d}\over{a}}\).................(iii)

On checking equation (i):=> \(2 + 1 + 1\)
=>\(4\)
=> \({{-(-4)}\over{1}} = {{-b}\over{a}}\)

On checking equation (ii):=> \(2 × 1 + 1 × 1 + 1 × 2\)

=>\(2 + 1 + 2 = 5\)
=> \({{5}\over{1}} = {{c}\over{a}}\)
On checking equation (iii):=> \(2 × 1 × 1\)
=>\(2\)
=> \( {{-(-2)}\over{1}}= {{-d}\over{a}}\)

Hence it is verified the numbers given alongside of the cubic polynomials are their zeroes.