Question

One zero of a quadratic polynomial is twice the other. If the sum of zeroes is (–6), find the polynomial.

Answer 1

1. Identify the relationship between zeroes

Let the two zeroes of the polynomial be α and β. According to the problem, one zero is twice the other, which can be expressed as:

α = 2β

2. Solve for individual zeroes

The problem states that the sum of the zeroes is –6. Using the relationship from the previous step:

α + β = –6

(2β) + β = –6

3β = –6

β = –2

Now, find the first zero:

α = 2(–2)

α = –4

3. Calculate the product of zeroes

To form the polynomial, calculate the product of the zeroes (αβ):

Product = (–4) × (–2)

= 8

4. Form the quadratic polynomial

A quadratic polynomial with a given sum (S) and product (P) of zeroes is represented by the formula:

P(x) = k(x² – Sx + P)

Substituting the known values (S = –6 and P = 8):

P(x) = k(x² – (–6)x + 8)

P(x) = k(x² + 6x + 8)

Taking k = 1 for simplicity, the standard polynomial is x² + 6x + 8.