# 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:

• 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.

## 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.