1.Find a quadratic polynomial whose sum and product of zeroes are √2 and 3 respectively. 2. If m and n are zeroes of the polynomial 3x²+11x-4, find the value of m/n+n/m. 3.If p and q are zeroes of the polynomial t²-4t+3,show that 1/p+ 1/q -2pq+14/3=0. 4.Obtain all zeroes of the polynomial f(x)=2x⁴+x³-14x²-19x-6, if two of its zeroes are -2 and -1. 5. Find all the zeroes of the polynomial x⁴-3x³+6x-4, if two of its Zeroes are√2 and -√2. 6. Find all the zeroes of the polynomial 2x⁴-9x³+5x²+3x-1,if two of its zeroes are 2+√3 and 2- √3. 7. Find all the zeroes of the polynomial x³- 4x²- 3x- 1,if two of its zeroes are √3 and - √3. 8. Find all the zeroes of the polynomial 2x³+ x²- 6x- 3, if two of its zeroes are √3 and - √3. 9. Find the zeroes of a polynomial x³-5x²-16x+80, if its zeros are equal in magnitude but opposite in s
Answers
Solutions:-
1.
A quadratic equation in 'x' can be written as
x² - (sum of roots)x + (product of roots) = 0
→ x²- √2x + 3 = 0
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2.
Sum of roots = m+n = -11/3
Product of roots = mn = -4/3
m/n + n/m
= (m²+n²)/mn
= {(m+n)²-2mn}/mn
= -145/12
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3.
Sum of roots = p+q = 4
product of roots = pq = 3
1/p+1/q-2pq+14/3
= p+q/pq -2pq + 14/3
= 0
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4.
Let the other two zeroes of polynomial be a,b.
Then all zeroes are a,b,-2,-1
Sum of zeroes = a+b-3 = -1/2
Product of zeroes = 2ab = -3
Solve these both equations.
You get a= 3, b = -1/2
So, all the zeroes of given polynomial are -2,-1,3,-1/2
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5.
Same model as 4th question.
6.
Same model as 4th question.
7.
Same model as 4th question.
8.
Same model as 4th question.
9.
Same model as 4th question.