H = #0#2 #1 2 



G = # 2 #3 - 3#o#i#2 

 F = #o 3 #4 4#o 2 #i#3 



ALGEBRA 

 2#i 3 



- 6# #i 2 #2 - 3#i 4 



F = # 2 / - 



/ = flo#2#4 + 2#i#2#3 #o#3 2 ~~ #1 2 #4 



A = / 3 - 27/ 2 = the discriminant 

 G 2 + 4# 3 = # 2 (#/ - #o/). 



II 



Nature of the roots of the biquadratic: 



A = o Equal roots are present 



Two roots only equal: 7 and / are not both zero 



Three roots are equal : I = / = o 



Two distinct pairs of equal roots: G = o; a 2 / - i2# 2 = o 



Four roots equal : H = I = J = o. 

 A < o Two real and two complex roots 

 A > o Roots are either all real or all complex: 



H < o and a^I i2H 2 < o Roots all real 



H > o and a Q 2 I i2# 2 > o Roots all complex. 



DETERMINANTS 



1.300 A determinant of the nth order, with n 2 elements, is written: 



1.301 A determinant is not changed in value by writing rows for columns and 

 columns for rows. 



1.302 If two columns or two rows of a determinant are interchanged the re- 

 sulting determinant is unchanged in value but is of the opposite sign. 



1.303 A determinant vanishes if it has two equal columns or two equal rows. 



1.304 If each element of a row or a column is multiplied by the same factor 

 the determinant itself is multiplied by that factor. 



