Question

If α and β are the zeros of polynomial x2 + 3x - 4, find a polynomial whose zeros are 3α+1

and 3β+1​

Answer 1

Step-by-step explanation:

$given \: polynomial \: is \: {x}^{2} + 3x - 4 \\ comparing \: it \: with \: a {x}^{2} + bx + c \: we \: find \: a = 1 \: b = 3 \: and \: c = - 4 \\ \alpha \: and \: \beta \: are \: the \: zeros \: of \: given \: polynomial \\ \alpha + \beta = - \frac{b}{a} = - \frac{3}{1} = - 3....(i) \\ \alpha \beta = \frac{c}{a} = - 4 \\ now \: \\ ({ \alpha - \beta })^{2} = ({ \alpha + \beta })^{2} - 4 \alpha \beta \\ = ({ - 3})^{2} - 4( - 4) = 9 + 16 = 25 \\ \alpha - \beta = 5....(ii) \\ solving \: equtions \: (i) \: and \: (ii) \: we \: find \\ \alpha = 1 \\ \beta = - 4 \\now \: \\ 3 \alpha + 1 = 4 \\ 3 \beta + 1 = - 11 \\ (3 \alpha + 1) + (3 \beta + 1) = 4 - 11 = - 7 \\ (3 \alpha + 1) . (3 \beta + 1) = 4.( - 11) = - 44 \\ required \: polynomial \: is \: \\ {x}^{2} -( (3 \alpha + 1) + (3 \beta + 1)) x + (3 \alpha + 1) . (3 \beta + 1) \\ = {x}^{2} - ( - 7)x + ( - 44) \\ = {x}^{2} + 7x - 44$