Question

if alpha and beta are the zeros of 3 x square - 7 x minus 6 find a polynomial whose zeros are twice alpha + thries beta and thries alpha + twice beta​

Answer 1

Answer:-

Given Polynomial => 3x² - 7x - 6 = 0

By splitting the middle term,

3x² - 9x + 2x - 6 = 0

3x(x - 3) + 2(x - 3) = 0

(3x + 2)(x - 3) = 0

3x + 2 = 0

3x = - 2

x = - 2/3

x - 3 = 0

x = 3

$Therefore, \: \alpha = \frac{ - 2}{3} \: and \: \beta = 3.$

$2 \alpha + 3 \beta \: and \: 3 \alpha + 2 \beta \: are \: the \: roots \: of \: new \: polynomial. \\ \\ = > \: 2 \alpha + 3 \beta = 2( \frac{ - 2}{3} ) + 3(3) \\ \\ 2 \alpha + 3 \beta = \frac{ - 4 + 27}{3} = \frac{23}{3} \\ \\ = > 3 \alpha + 2 \beta = 3( \frac{ - 2}{3} ) + 2(3) \\ \\ 3 \alpha + 2 \beta = \frac{ - 6 + 18}{3} = \frac{12}{3} = 4$

Sum of the roots = 23/3 + 4 = 35/3

Product of the roots = (23/3)(4) = 92/3

We know that,

General form of a quadratic equation = x² - Sum of the roots)x + product of the roots = 0

New polynomial = x² - (35/3)x + 92/3 = 0

Taking LCM ,

(3x² - 35x + 92)/3 = 0

3x² - 35x + 92 = 0

Hence, the new quadratic equation formed is 3x² - 35x + 92 = 0.