Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively (i) $\frac{1}{4}, -1$ (ii) $\sqrt{2}, \frac{1}{4}$ (iii) $0, \sqrt{5}$ (iv) $1, 1$ (v) $- \frac{1}{4}, \frac{1}{4}$ (iv) $4, 1$

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively

(i) $\frac{1}{4}, -1$

(ii) $\sqrt{2}, \frac{1}{4}$

(iii) $0, \sqrt{5}$

(iv) $1, 1$

(v) $- \frac{1}{4}, \frac{1}{4}$

(iv) $4, 1$

 

The quadratic polynomial with given sum and product of zeroes can be found using the standard form:

$x^2-(\text{sum of zeroes})x+(\text{product of zeroes})=0$

(i) Given sum = $\frac{1}{4}$, product = $-1$

$x^2-\frac{1}{4}x-1=0$

(ii) Given sum = $\sqrt{2}$, product = $\frac{1}{4}$

$x^2-\sqrt{2}x+\frac{1}{4}=0$

(iii) Given sum = $0$, product = $\sqrt{5}$

$x^2+\sqrt{5}=0$

(iv) Given sum = $1$, product = $1$

$x^2-x+1=0$

(v) Given sum = $-\frac{1}{4}$, product = $\frac{1}{4}$

$x^2+\frac{1}{4}x+\frac{1}{4}=0$

(vi) Given sum = $4$, product = $1$

$x^2-4x+1=0$

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