Find a quadratic polynomial, the sum and product of whose zeroes are $-\dfrac{1}{4}$ and $\dfrac{1}{4}$ respectively.
Step-by-Step Solution
Key Concept: Quadratic polynomial $= k[x^2 - (\text{sum})x + (\text{product})]$.
Stepwise Solution:
Let sum $S = -1/4$ and product $P = 1/4$. [0.5 Mark]
Required polynomial $p(x) = k \left[ x^2 - S x + P \right] = k \left[ x^2 - \left(-\dfrac{1}{4}\right)x + \dfrac{1}{4} \right] = k \left[ x^2 + \dfrac{1}{4}x + \dfrac{1}{4} \right]$. [1.0 Mark]
Taking $k = 4$, we get $p(x) = 4x^2 + x + 1$. [0.5 Mark]
Marking Scheme:
• Identifying sum and product values: 0.5 Mark
• Substituting into general polynomial formula: 1.0 Mark
• Simplifying to integer coefficients ($4x^2 + x + 1$): 0.5 Mark
Correct Answer: