Question

if α and β are the zeros of the quadratic polynomial f(x) = x^2-1 ,find a quadratic polynomial whose zeros are 2α/β and 2β/α

Answer 1

Answer:

The required polynomial is $x^2+4x+4$

Step-by-step explanation:

f(x) = x² - 1

f(x) = (x+1)(x-1)

clearly, f(1)=0 and f(-1)=0

Then 1 and -1 ae zeros of f(x)

That is

$\alpha=1\:and\:\beta=-1$

$\frac{2\alpha}{\beta}\\\\=\frac{2(1)}{(-1)} \\\\=-2$

$\frac{2\beta}{\alpha}\\\\=\frac{2(-1)}{1} \\\\=-2$

$\frac{2\alpha}{\beta}+\frac{2\beta}{\alpha}$

=-2-2

=-4

$\frac{2\alpha}{\beta}.\frac{2\beta}{\alpha}$

=(-2)(-2)

=4

The required quadratic polynomial is

$x^2-(\frac{2\alpha}{\beta}+\frac{2\beta}{\alpha})x+(\frac{2\alpha}{\beta}.\frac{2\beta}{\alpha})\\\\=x^2-(-4)x+4\\\\=x^2+4x+4$

Answer 2

Answer:

The required polynomial is

Step-by-step explanation:

f(x) = x² - 1

f(x) = (x+1)(x-1)

clearly, f(1)=0 and f(-1)=0

Then 1 and -1 ae zeros of f(x)

That is

=-2-2

=-4

=(-2)(-2)

=4

The required quadratic polynomial is

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