Question

if a,b are zeroes of polynomial x^2 -5x + 6,then the polynomial having a+b ,a-b is.​

Answer 1

Given:

A polynomial x²-5x+6 having zeroes a and b

To Find:

A polynomial whose zeroes are a+b and a-b

Solution:

Let given polynomial be p(x)=x²-5x+6

We know that,

A polynomial in the form of ax²+bx+c can be expressed as

$\sf{x^{2}-(Sum\:of\:Zeroes)x+(Product\:of\:Zeroes)}$

Where,

$\sf{Sum\:of\:Zeroes=\dfrac{-Coefficient\:of\:x}{Coefficient\:of\:x^{2}}=\dfrac{-b}{a}}$

$\sf{Product\:of\:Zeroes=\dfrac{Constant\:Term}{Coefficient\:of\:x^{2}}=\dfrac{c}{a}}$

Also,

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

Now,

In given polynomial

$\longrightarrow\sf{Sum\:of\:Zeroes=\dfrac{-(-5)}{1}}$

$\longrightarrow\sf{a+b=5}----------(1)$

$\longrightarrow\sf{Product\:of\:Zeroes==\dfrac{6}{1}}$

$\longrightarrow\sf{ab=6}-----------(2)$

Now, the required polynomial will be

$\sf{x^{2}-(a+b+a-b)x+(a+b)(a-b)}$

$\sf{x^{2}-(5+\sqrt{(a+b)^{2}-4ab})x+(5)(\sqrt{(a+b)^{2}-4ab})}$

From (1) and (2),

$\sf{x^{2}-(5+\sqrt{(5)^{2}-4(6)})x+(5)(\sqrt{(5)^{2}-4(6)})}$

$\sf{x^{2}-(5+\sqrt{25-24})x+(5)(\sqrt{25-24})}$

$\sf{x^{2}-(5+\sqrt{1})x+(5)(\sqrt{1})}$

$\sf{x^{2}-(5+1)x+5}$

$\sf\pink{x^{2}-6x+5}$

Hence, the required polynomial is $\sf\green{x^{2}-6x+5}$.