Question

If alpha and beta are the zeros of the polynomial 2x^2-5x+7, then find the polynomial whose zeros are 2alpha+3beta and 3alpha+2beta

Answer 1

HI !

NOTE :-
 α² + β² can be written as (α + β)² - 2αβ

p(x) = 2x² - 5x + 7
a = 2 , b = - 5 , c = 7

α and β are the zeros of p(x)

we know that ,
sum of zeros = α + β
                      = -b/a
                      = 5/2

product of zeros = c/a
                           = 7/2
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2α + 3β and 3α + 2β are zeros of a polynomial.

sum of zeros = 2α + 3β+ 3α + 2β
                      = 5α + 5β
                      = 5 [ α + β]
                     = 5 × 5/2
                    = 25/2

product of zeros = (2α + 3β)(3α + 2β)
                          = 2α [ 3α + 2β] + 3β [3α + 2β]
                         = 6α² + 4αβ + 9αβ + 6β²
                         = 6α² + 13αβ +  6β²
                         = 6 [ α² + β² ] + 13αβ
                         = 6 [ (α + β)² - 2αβ ] + 13αβ
                         = 6 [ ( 5/2)² - 2 × 7/2 ] + 13× 7/2
                         = 6 [ 25/4 - 7 ] + 91/2
                         = 6 [ 25/4 - 28/4 ] + 91/2
                         = 6 [ -3/4 ] + 91/2
                        = -18/4 + 91/2
                        = -9/2 + 91/2
                        = 82/2
                        = 41

                                                                           -18/4 = -9/2 [ simplest form ]

a quadratic polynomial is given by :-

k { x² - (sum of zeros)x + (product of zeros) }

k {x² - 5/2x + 41}

k = 2

2 {x² - 5/2x + 41 ]

2x² - 5x + 82                   -----> is the required polynomial

Answer 2

Answer:

The polynomial whose zeros are 2α+3β and 3α+2β is 2x² -25x +82

Step-by-step explanation:

Given,

α and β are the zeros of 2x² -5x+7

To find,

The polynomial whose zeros are 2α+3β and 3α+2β

Recall the concepts

If α and β are the roots of the quadratic equation of ax²+bx+c = 0, then the sum of roots  = α + β = $\frac{-b}{a}$ and the product of root = αβ = $\frac{c}{a}$

If the roots of the quadratic equation is given then, we can form the quadratic equation by the formula

x² - (sum of roots )x+ product of roots= 0, ---------------------(A)

Solution:

Since α and β are the zeros of 2x² -5x+7, then

α + β = $\frac{-b}{a}$  = $\frac{5}{2}$ and αβ =  $\frac{c}{a}$ = $\frac{7}{2}$

α + β = $\frac{5}{2}$ and αβ = $\frac{7}{2}$ ----------------------(1)

To find the polynomial whose zeros are 2α+3β and 3α+2β

Sum of zeros = 2α+3β+ 3α+2β = 5α+5β = 5(α+β) = 5× $\frac{5}{2}$ =$\frac{25}{2}$ (from(1))

Sum of zeros  =$\frac{25}{2}$

Product of zeros = (2α+3β)(3α+2β)

= 6α²+ 4αβ+9αβ+6β²

= 6(α²+β²) +13αβ

= 6((α+β)² - 2αβ) + 13αβ

= 6(α+β)² - 12αβ + 13αβ

= 6(α+β)² + αβ

= 6× $\frac{25}{4} +\frac{7}{2}$

= $\frac{75}{2}$ + $\frac{7}{2}$

= $\frac{82}{2}$

= 41

Product of zeros = 41

The required equation is

x² - (sum of roots )x+ product of roots= 0

x² - ($\frac{25}{2}$ )x+ 41= 0

2x² -25x +82 = 0

∴The polynomial whose zeros are 2α+3β and 3α+2β is 2x² -25x +82

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