Question

Find the zeroes of the polynomial \(r(x) = 4x^2 + 3x - 1\). Hence, write a polynomial whose zeroes are reciprocal of the zeroes of \(r(x)\).

Hint:

To get a polynomial with reciprocal roots, take \(x = \frac{1}{\alpha}, \frac{1}{\beta}\) and multiply \( (x - \frac{1}{\alpha})(x - \frac{1}{\beta}) \) or invert roots and form new factors.

Answer 1

Given: \[ r(x) = 4x^2 + 3x - 1 \] Use quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{9 + 16}}{8} = \frac{-3 \pm \sqrt{25}}{8} = \frac{-3 \pm 5}{8} \] \[ \text{Zeroes: } x = \frac{1}{4},\ -1 \] Reciprocal of zeroes: \(4,\ -1\) So, required polynomial = \[ (x - 4)(x + 1) = x^2 - 3x - 4 \] \[ \boxed{\text{Polynomial: } x^2 - 3x - 4} \]