If α, β are the zeros of the quadratic polynomial f(x) = 2x2 - 5x + 7. Find a polynomial whose zeros are 2α + 3β and 3α + 2β.

By Ritesh|Updated : November 9th, 2022

The polynomial whose zeros are 2α + 3β and 3α + 2β is 2x2 - 25 + 82. Steps to Find a polynomial whose zeros are 2α + 3β and 3α + 2β:

Given that p(x) = 2x2 - 5x + 7 and its zeros are denoted by α, β

Now we have to compare p(x) = 2x2 - 5x + 7 with ax2 + bx + c

a = 2, b = -5 and c =7

we know that,

sum of zeros = α + β = -b/a

Substituting the values we get,

= 5/2

product of zeros = c/a

Substituting the values we get:

= 7/2

2α + 3β and 3α + 2β are the zeros of the required polynomial,

sum of zeros = 2α + 3β + 3α + 2β

= 5α + 5β

Taking common,

= 5 (α + β)

= 5 x 5/2

Substituting the values we get:

= 25/2

product of zeros = (2α + 3β) (3α + 2β)

= 2α (3α + 2β) + 3β (3α + 2β)

On multiplying the above equation we get:

= 6α2 + 4αβ + 9αβ + 6β2

On simplifying:

= 6α2 + 13αβ + 6β2

= 6 [α2 + β2] + 13αβ

= 6 [(α + β)2 - 2αβ] + 13αβ

Substituting the values we get:

= 6 [(5/2)2 - 2 x 7/2] + 13 x 7/2

= 6 [25/4 - 7] + 91/2

= 6 [25 - 28/4] + 91/2

= 6 [-¾] + 91/2

= -18/4 + 91/2

= -9/2 + 91/2

= 82/2

= 41

A quadratic polynomial is given by:

k [x2 - (sum of zeros)x + (product of zeros)]

= k [x2 - 25x/2 + 41]

Take k = 2

= 2 [x2 - 25x/2 + 41]

= 2x2 - 25 + 82 is the required polynomial.

Summary:

If α, β are the zeros of the quadratic polynomial f(x) = 2x2 - 5x + 7. Find a polynomial whose zeros are 2α + 3β and 3α + 2β.

If α and β are the zeros of the quadratic polynomial f(x) = 2x2 - 5x + 7. The polynomial whose zeros are 2α + 3β and 3α + 2β is 2x2 - 25 + 82.

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