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6. p, q, r are given real numbers. Consider the polynomials (q-r)(x-q)(x-r) + (r-p)(x-r)(x-p) + +(p-q)(x-p)(x-q) + k for all real k. Then [ ] A) there exists a k such that the degree of the polynomial is 2 B) there exists a k such that the degree of the polynomial is 1 C) there exists no k such that the degree of the polynomial is 0 D) there exists no k such that the degree of the polynomial is undefined E) none of these sir plzzzzzzzz solve .... i will surely approve.....

6. p, q, r are given real numbers. Consider the polynomials (q-r)(x-q)(x-r) + (r-p)(x-r)(x-p) + +(p-q)(x-p)(x-q) + k for all real k. Then [ ]


A) there exists a k such that the degree of the polynomial is 2


B) there exists a k such that the degree of the polynomial is 1


C) there exists no k such that the degree of the polynomial is 0


D) there exists no k such that the degree of the polynomial is undefined


E) none of these


 


sir plzzzzzzzz      solve    .... i will surely approve.....


 

Grade:Upto college level

1 Answers

Pushkar Pandit
31 Points
10 years ago

(q-r)(x-q)(x-r) + (r-p)(x-r)(x-p) + (p-q)(x-p)(x-q) + k 

-->first let us solve 

(q-r)(x-q)(x-r)

we will get,

qx^2-q^2x-rx^2+rqx-rqx+rq^2+r^2x-r^2q

x^2(q-r)+(r^2-q^2)x+rq(q-r)

Now solve

(r-p)(x-r)(x-p)

 

Here we can see the similar pattern i.e.Herein place of "q" it is "r" and in place of "r" it is "p".So,we can write

x^2(r-p)+(p^2-r^2)x+rp(r-p)

Similarly,3rd erm also folows similar pattern So

x^2(p-q)+(q^2-p^2)x+qp(p-q)

When we will add all the 3 terms it will sum out to 0and only 'k' is left

As 'k' is a constant so the possible degree of polynomial is  0

It is satisfying only fifth option so answer is none of these 

 

 


 

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