Question

if alpha and beta are the two zeroes of polynomial 25p^2-15p+2, find a quadratic polynomial whose zeroes are 1\2alpha and 1\2beta

Answer 1

Answer:

Required polynomial = 8p²-30p+25

Explanation:

Given $\alpha\: and \: \beta$ are two zeroes of polynomial 25p²-15p+2 ,

Compare this with ap²+bp-+c, we get

a = 25, b = -15, c = 2

i ) sum of the zeroes = -b/a

$\alpha + \beta = \frac{-(-15)}{25}$

= $\frac{3}{5}$ ----(1)

ii) product of the zeroes = c/a

$\alpha\beta=\frac{2}{25}$ ---(2)

Now ,

iii) If zeroes of the polyomial

are $\frac{1}{2\alpha}\: and \frac{1}{2\beta}$

iv ) Sum of the zeroes of the polyomial = $\frac{1}{2\alpha}+\frac{1}{2\beta}$

= $\frac{1}{2}\times \frac{\beta+\alpha}{\alpha\beta}$

= $\frac{1}{2}\times\frac{\frac{3}{5}}{\frac{2}{25}}$

= $\frac{15}{4}$ ---(3)

v) Product of the zeroes

= $\frac{1}{2\alpha}\frac{1}{2\beta}$

=$\frac{1}{4\alpha\beta}$

= $\frac{1}{4\times\frac{2}{25}}$

= $\frac{25}{8}$ ----(4)

______________________

Form of a quadratic polynomial

= k[p²-(sum of the zeroes)p+(product of the zeroes)

________________________

= k[p²-(15/4)p(25/8)]

For all real values of k it is true .

If k = 8 then

Required polynomial

= 8p²-30p+25

Therefore,

Required polynomial = 8p²-30p+25

••••

Answer 2

Answer:8p2-30p+25

Step-by-step explanation in picture