ЛЕКЦІЇ ² ВПРАВИ

14.1.6. Ïðîïîðö³¿
a c
= ⇔ ad = bc .
b d
(a, d — êðàéí³ ÷ëåíè ïðîïîðö³¿, b, c — ñåðåäí³ ÷ëåíè ïðîïîðö³¿)
òà ¿õí³ âëàñòèâîñò³:
a+ b c+ d a−b c− d a+ b c+ d a−b c−d
= ; = ; = ; = ;
a c a c b d b d
a a
ab = a b ; ( b );
b
= b
≠0
a+ b ≤ a + b ; a−b ≥ a − b .
14.2.2. Ôîðìóëè ñêîðî÷åíîãî ìíîæåííÿ:
( )
2 2 2
a± b = a ± 2 ab+
b ;
a = c ; a = c ; b = d ;
b =
d
a+ b c+ d a−b c− d a+ b c+ d a−b c−d
a+ b c+ d a+ c a c a+
b a b
= ; = = ; = = ;
a−b c− d b+ d b d c+
d c d
a−c a c a−b a b
= = ; = = .
b−d b d c−d c d
Çàãàëüí³ ïîõ³äí³ ïðîïîðö³¿:
ma + nb mc + nd ma + nc mb + nd
= ; =
ma+ nb mc+ nd ma+ nc mb+
nd
1 1 1 1 1 1 1 1
äå m, n, m 1 , n 1 — äîâ³ëüí³ ÷èñëà.
14.2. ÀËÃÅÁÐÀ
;
( )
( )
3 3 2 2 3
a± b = a ± 3a b+ 3 ab ± b ;
( )( )
2 2
a − b = a− b a+
b ;
( )( )
3 3 2 2
a ± b = a± b a m ab+
b ;
2 2 2 2
a + b + c = a + b + c + 2ab + 2ac + 2 bc .
14.2.3. Ñòåïåí³ òà ¿õí³ âëàñòèâîñò³:
( )
n
0 1
a = a14243 ⋅a⋅K⋅ a; a = 1 a ≠ 0 ; a = a
;
1
n
n
m
n
1
n n m −n
a = a; a = a ; a = ( m,
n∈N n
);
a
14.2.1. Ìîäóëü ä³éñíîãî ÷èñëà ³ éîãî âëàñòèâîñò³:
p p p p
q
pq p q p
( ) = ; ( ) = ; = + q
( , ∈ )
ab a b a a a a a p q R ;
a
⎧ a, a ≥ 0,
= ⎨
⎩ − a, a < 0;
p p p
⎛a⎞
a a p−q
⎜ ⎟ = ; = a ( p, q∈R , b ≠0
p q
).
⎝b⎠
b a
a ≥ 0; a = 0⇔ a = 0;
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