Question

if alpha and beta are the zeroes if the quadratic polynomial f(x)=
$ax ^{2} + bx + c$
then evaluate ...
1)
$\alpha ^{4} + \beta ^{4}$

2)
$\alpha ^{2} \div \beta ^{2} + \beta ^{2} \div \alpha ^{2}$


​

Answer 1

$GOOD \: \: MORNING \: \\ \\ f(x) = ax {}^{2} + bx + c \\ \\ \alpha + \beta = - b \div a \\ and \: \: \\ \alpha \times \beta = c \div a \\ \\ \alpha {}^{2} + \beta {}^{2} = ( \alpha {}^{2} + \beta ) {}^{2} - 2( \alpha \beta ) \\ \alpha {}^{2} + \beta {}^{2} = (b \div a) {}^{2} - 2(c \div a) \\ \alpha {}^{2} + \beta {}^{2} = (b {}^{2} - 2ac) \div a {}^{2} \\ 1) \: \: \: \: \: \\ \alpha {}^{4} + \beta {}^{4} =( \alpha {}^{2} + \beta {}^{2} ) {}^{2} - 2( \alpha \beta ) {}^{2} \\ \\ \alpha {}^{4} + \beta {}^{4} = (( b {}^{2} - 2ac) \div 2a) {}^{2} - 2c {}^{2} \div a {}^{2}$