Question

разложите на множители многочлен х^4-25х^2+144

Answer 1

$x^4-25x^2+144 = (x^2)^2-2cdotfrac{25}{2}x^2+(frac{25}{2})^2-frac{625}{4}+144 = \ = (x^2-frac{25}{2})^2-frac{49}{4} = (x^2-frac{25}{2})^2-(frac{7}{2})^2 = (x^2-frac{25}{2} - frac{7}{2})(x^2-frac{25}{2} + frac{7}{2}) = \ =(x^2-frac{32}{2})(x^2-frac{18}{2}) = (x^2-16)(x^2-9) = \ =(x-4)(x+4)(x-3)(x+3).$
II
$x^4-25x^2+144=0, \ a=x^2, a^2-25a+144=0, \ a_1=9, a_2=16, \ a^2-25a+144=(a-9)(a-16), \ x^4-25x^2+144=(x^2-9)(x^2-16)=(x-3)(x+3)(x-4)(x+4).$