Question

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

Hint:

For polynomial \(x^2 + bx + c\), the sum of zeroes is \(-b\) and the product is \(c\). This shortcut only works when the coefficient of \(x^2\) is 1.

Answer 1

Step 1: Understanding the Concept:
A quadratic polynomial \(p(x)\) with zeroes \(\alpha\) and \(\beta\) can be written as \(k[x^2 - (\alpha + \beta)x + \alpha\beta]\).
Step 2: Key Formula or Approach:
1. Let the zeroes be \(\alpha\) and \(2\alpha\).
2. Sum of zeroes: \(\alpha + \beta = -b/a\)
3. Product of zeroes: \(\alpha\beta = c/a\)
Step 3: Detailed Explanation:
1. Given: Sum of zeroes = \(-6\).
\[ \alpha + 2\alpha = -6 \implies 3\alpha = -6 \implies \alpha = -2 \] 2. The zeroes are \(-2\) and \(2(-2) = -4\).
3. Product of zeroes:
\[ \alpha\beta = (-2) \times (-4) = 8 \] 4. Construct the polynomial:
\[ p(x) = x^2 - (\text{Sum})x + (\text{Product}) \] \[ p(x) = x^2 - (-6)x + 8 = x^2 + 6x + 8 \]
Step 4: Final Answer:
The polynomial is \(x^2 + 6x + 8\).