Question

If α,β are the polynomial x^2-2x-15 then form a quadratic polynomial whose zeroes are (2α)and (2β).

X^2-4x-60

Answer 1

${x}^{2} - 2x - 15 \\ {x}^{2} -5x + 3x - 15 \\ x(x - 5) + 3(x - 5) \\ (x - 5)(x + 3) \\ x = - 5 \: and \: 3 \\ let \: \alpha = - 5 \: and \: \beta = 3 \\ \\ \alpha + \beta = - 5 + 3 = - 2 \\ \alpha \beta = - 5 \times 3 = - 15 \\ \\ polynomial \: whose \: zeroes \: are \: 2 \alpha \: and \: 2 \beta \\ = k( {x}^{2} - (2 \alpha + 2 \beta )x + 2 \alpha 2 \beta \\ = k( {x}^{2} - 2( \alpha + \beta )x + 2( \alpha \beta )) \\ = k( {x}^{2} - 2( - 2)x + 2( - 15)) \\ = k( {x}^{2} + 4x - 15)$