Form a quadratic polynomial whose zeroes are : (2+1/√2) and (2-1/√2)
Answers
EXPLANATION.
Quadratic polynomial.
Whose zeroes are = (2 + 1/√2) and (2 - 1/√2).
As we know that,
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = - b/a.
⇒ 2 + 1/√2 + 2 - 1/√2 = 4.
⇒ α + β = 4.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
⇒ (2 + 1/√2) x (2 - 1/√2).
As we know that,
Formula of :
⇒ (x² - y²) = (x + y)(x - y).
⇒ (2)² - (1/√2)².
⇒ 4 - 1/2.
⇒ (8 - 1)/2 = 7/2.
⇒ αβ = 7/2.
As we know that,
Formula of quadratic polynomial.
⇒ x² - (α + β)x + αβ.
Put the values in the equation, we get.
⇒ x² - (4)x + (7/2) = 0.
⇒ x² - 4x + 7/2 = 0.
⇒ 2x² - 8x + 7 = 0.
MORE INFORMATION.
Conjugate roots.
(1) = If D < 0.
One roots = α + iβ.
Other roots = α - iβ.
(2) = If D > 0.
One roots = α + √β.
Other roots = α - √β.
Given : Zeroes of a quadratic polynomial are (2 + 1/√2) and (2 - 1/√2)
To find : Quadratic polynomial
Solution :
A quadratic polynomial is a polynomial function with degree 2. Degree is the highest power.
General form of quadratic polynomial :-
- x² - (sum of zeroes) x + Product of zeroes
So, inorder to form a quadratic polynomial, we have to find the value of sum and product of zeroes.
• Finding sum of zeroes :-
➝ (2 + 1/√2) + (2 - 1/√2 )
➝ 2 + 1/√2 + 2 - 1/√2
➝ 2 + 2 + 1/√2 - 1/√2
➝ 4
• Finding product of zeroes :-
➝ ( 2 + 1/√2 ) ( 2 - 1/√2 )
Apply algebraic identity : ( A+B )(A-B) = A² - B²
➝ ( 2 )² - ( 1 / √2 )²
➝ 4 - 1 / 2
➝ ( 8 - 1 ) / 2
➝ 7 / 2
Since we have obtained the sum and product of zeroes, now we can easily find the quadratic polynomial by substituting them in general form of quadratic polynomial.
• Finding quadratic polynomial :-
➝ x² - ( sum of zeroes ) x + Product
➝ x² - ( 4 ) x + 7 / 2
➝ x² - 4x + 7/2
By multiplying by 2,
➝ 2x² - 8x + 7
This is the required polynomial.
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