If α and β are the Zeros of the Polynomial x^2-2x-15, Then Form a Quadratic Polynomial whose Zeros are 2α and 2β
By BYJU'S Exam Prep
Updated on: October 17th, 2023
x^2 – 4x – 60 is the polynomial.
Let us see the steps to reach to the result quickly:
- Start with the original polynomial: x^2 – 2x – 15.
- We know that if α and β are zeros of the polynomial, then (x – α) and (x – β) are factors of the polynomial.
- To find a quadratic polynomial with zeros 2α and 2β, we need to double the zeros. Therefore, the new zeros will be 2α and 2β.
- Construct the quadratic polynomial using the new zeros: (x – 2α)(x – 2β) = 0.
- Expand the equation: x^2 – 2αx – 2βx + 4αβ = 0.
- Determine the values of (2α + 2β) and 4αβ using the original polynomial.
Table of content
If α and β are the Zeros of the Polynomial x²-2x-15, Then Form a Quadratic Polynomial whose Zeros are 2α and 2β
Solution:
Given the polynomial x² – 2x – 15 with zeros α and β, we want to find a quadratic polynomial with zeros 2α and 2β.
To do this, we can simply multiply the original polynomial by (x – 2α)(x – 2β). This is because when we multiply a polynomial by its factors, the resulting polynomial will have those values as zeros.
So, we have:
(x – 2α)(x – 2β) = 0
Expanding this equation, we get:
x² – 2βx – 2αx + 4αβ = 0
Combining like terms, we have:
x² – (2β + 2α)x + 4αβ = 0
Now, we know that the original polynomial x² – 2x – 15 has α and β as its zeros.
This means that:
sum of the zeros = -b/a (-(-2)) / 1 = 2
Product of the zeros = c/a = (-15) / 1 = -15
Using this information, we can simplify the equation further:
x² – (2β + 2α)x + 4αβ = 0
x² – 2(α + β)x + 4αβ = 0
x² – 2(2)x + 4(-15) = 0
x² – 4x – 60 = 0
Therefore, the quadratic polynomial with zeros 2α and 2β is x² – 4x – 60.
Answer:
Quadratic Polynomial whose Zeros are 2α and 2β is x² – 4x – 60
Similar Questions:
- α, β and γ are the Zeroes of Cubic Polynomial P(x) = ax^3 + bx2 + cx + d, (a≠0). Then Product of their Zeroes [αβγ] is?
- P(x)=x2 + 2(√2)x – 6 Find the Zero of the Polynomial and Verify between Zero and Coefficients
- Find the Quadratic Polynomial Whose Zeros are 3+√5 and 3-√5
- Find the Zeros of Polynomial p(x)=4x²-4x+1 and Verify the Relationship between the Zeros and the Coefficients
- If α+β=24 and α-β=8. Find the Polynomial having Alpha and Beta as its Zeros. Also Verify the Relationship between the Zeros and the Coefficients of the Polynomial
- Find the Zeroes of the Polynomial x² + x – p(p+1)
- A Quadratic Polynomial, the Sum of Whose Zeros is 0 and One Zero is 3, is
- Find the Zeros of the Quadratic Polynomial √3x² – 8x + 4√3
- Find the Quadratic Polynomial, Sum of Whose Zeros is (5/2) and Their Product is 1. Hence, Find the Zeros of the Polynomial
