06. Find a quadratic polynomial whose zeroes are 5 + √2 and 5 - √2.
Answers
Heya!!
let the Quadratic polynomial be F(x)
F(x) = x² - (Sum of zeros)x + (product of zeros)
Here, Sum of zeros = 5 + √2 + 5 -√2
Sum of zeros = 10
And
product of zeros = (5 + √2 ) (5 -√2)
product of zeros = (5)² - (√2)²
product of zeros = 25 - 2 = 23
So, Quadratic polynomial is
F(x) = x² - (10)x + 23
F(x) = x² - 10x + 23
Note:-
F(x) =x² - (Sum of zeros)x + product of zeros. is the general formula to form a Quadratic polynomial when roots are given.
(a + b) (a - b) = a² - b²
Answer:
x² - 10x + 23
Step-by-step explanation:
Given that -
Roots of the quadratic equation are (5 + √2) and (5 - √2). Let us consider the roots of the equation be α and β. Thus,
α = (5 + √2) and β = (5 - √2),
Now, find the value of (α + β) -
α + β = 5 + √2 + 5 - √2
⇒ α + β = 5 + 5
⇒ α + β = 10
Also, find the value of (αβ) -
αβ = (5 + √2)(5 - √2)
⇒ αβ = (5)² - (√2)²
⇒ αβ = 25 - 2
⇒ αβ = 23
We know that;
Quadratic polynomial is given by-
= x² - (α + β)x + αβ
= x² - 10x + 23