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# 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.