Question

Sum of the two zeroes of a polynomial of degree 4 is - 1 and their product is - 2 . If other two zeroes are root 3 and - root 3, then find the polynomial

Answer 1

Sum of the two zeros of a polynomial of degree 4 is -1 and their product is -2. If the other two zeros are root3 and -root3, find the polynomial

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Answer 2

Answer:

Required polynomial p(x)=x⁴-3x²+x³-5x²-3x+6

Explanation:

Let 4 th degree polynomial be p(x)

It is given that sum of the two zeroes of a polynomial of degree 4 is -1 and their product is -2 .

Other two zeroes are √3 and -√3 .

i ) If we can find two numbers p and q such that

p+q = -1 ---(1)

and

pq = -2 ---(2)

then we can the factors.

(p-q)² = (p+q)²-4pq

= (-1)²-4(-2)

= 1+8

= 9

Now ,

p-q = √9

=> p-q = √3²

=> p-q = 3 ---(3)

Add equations (1) & (3) , we get

2p = 2

=> p = 1 ---(4)

Substitute p=1 in equation (2), we get

q = -2 ---(5)

Therefore,

1,-2 , √3 , -√3 are four factors of p(x)

=> p(x) = (x-1)(x+2)(x-√3)(x+√3)

= (x²+2x-x-2)[x²-(√3)²]

= (x²+x-2)(x²-3)

= x⁴-3x²+x³-3x-2x²+6

= x⁴+x³-5x²-3x+6

Therefore,

Required polynomial p(x)=x⁴-3x²+x³-5x²-3x+6

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