Question

Sum of zeroes of polynomial x2-ax+a is same as the product of zeros
True or false

Answer 1

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Answer 2

The given statement is true.

Given:

The statement 'Sum of zeroes of the polynomial (x² - ax + a) is the same as the product of zeros'.

To Find:

Whether the given statement is true or false.

Solution:

We have been given a quadratic equation (x² - ax + a)

The roots( or zeros) of the quadratic equation ax² + bx + c are given by the quadratic formula,

x =( -b ± $\sqrt{b^{2} - 4ac }$ )/2a

where

a = coefficient of x², b = coefficient of x, and c = constant

Hence, the two roots of the given quadratic equation (x² - ax + a) is

x = (a ± $\sqrt{a^{2} - 4a}$ )/ 2a

where  

coefficients of x² = 1

coefficients of x = -a  

constant value = a.

Sum of the roots of the given polynomial is

(a + $\sqrt{a^{2} - 4a}$ )/ 2a +  (a - $\sqrt{a^{2} - 4a}$ )/ 2

=  (a + $\sqrt{a^{2} - 4a}$  + a - $\sqrt{a^{2} - 4a}$ ) )/ 2 = 2a/2 = a

∴ Sum of the roots of the given polynomial is 'a'.   .............................(I)

We know that for a quadratic equation,

Product of its roots/zeros = (constant value) / (coefficient of x²)

Hence, for the given equation (x² - ax + a)

Product of its roots/zeroes = a/1 = a                       ...............................(II)

Hence, from equations (I) and (II)

The sum of roots of the given polynomial = Product of its roots = a

∴ The given statement is true.

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