17. Find the quadratic polynomial, the sum of whose zeros is 5/2 and their
product is 1. Hence, find the zeros of the polynomial.
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Answers
Step-by-step explanation:
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Given:-
- Sum of zeroes = 5/2
- Product of zeroes = 1
To Find:-
- The quadratic polynomial
Solution:-
Let α and β be the two zeroes of the required polynomial such that:-
- Sum of zeroes = α + β
- Product of zeroes = αβ
∵ Sum of zeroes and product of zeroes are (α + β) and (αβ) respectively, we can write:-
- α + β = 5/2 . . . (i)
- αβ = 1 . . . (ii)
We know,
A quadratic polynomial is in the form:-
- x² - (α + β)x + αβ . . . (iii)
Putting the values from equation (i) and (ii) into equation (ii), we get:-
x² - (5/2)x + 1
∴ The required quadratic polynomial is
- x² - 5/2x + 1
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Verification!!!
We got the quadratic polynomial as x² - 5/2x + 1
Let us find the zeroes of the polynomial.
p(x) x² - 5/2x + 1
Let us take the LCM as 2
= (2x² - 5x + 2)/2 = 0
By cross multiplying,
= 2x² - 5x + 2 = 0
By splitting the middle - term,
= 2x² - 4x - x + 2 = 0
Taking common terms,
= 2x(x - 2) - 1(x - 2) = 0
= (x - 2)(2x - 1) = 0
Either,
x - 2 = 0
⇒ x = 2
Or,
2x - 1 = 0
⇒ 2x = 1
⇒ x = 1/2
Let us verify the relation between the zeroes and the coefficients,
We know,
Sum of zeroes = -(Coefficient of x)/(Coefficient of x²)
Hence,
2 + 1/2 = -(-5/2)
= (4 + 1)/2 = 5/2
= 5/2 = 5/2
Also,
Product of zeroes = (Constant Term)/(Coefficient of x²)
= 2 × 1/2 = 1/1
= 1 = 1
Hence Verified!!!!
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