Page 8 - Algebra 1

Identities
An identity is an equality that is true for every variable. For example,
(x + 1) (x – 2) = x (x – 2) + 1(x – 2)
=x2 –2x+x–2=x2 –x–2 Therefore,(x+1)(x–2)=x2 –x–2isanidentity.
But,
x2 –x–2=4,forx=3
This is not an identity because it’s not true for every value of x.
REMEMBER:
(a+b)2 =a2 +2ab+b2
(a–b)2 =a2 –2ab+b2
a2 –b2 =(a+b)(a–b) (x+a)(x+b)=x2 +(a+b)x+ab
211212 x+2=(x+ )–2=(x– )+2
𝑥𝑥𝑥
(a+b+c)2 =a2 +b2 +c2+2(ab+bc+ca)
(a+b)3 =a3 +3a2b+3ab2 +b3
(a–b)3 =a3 –3a2b+3ab2 –b3
x3+y3 +z3 –3xyz=(x+y+z)(x2 +y2+z2 –xy–yz–zx)
Worked Example 5
2 Using the standard identities, find the value of 105 .
Solution:
105 can be written as (100 + 5)2 Usetheidentity,(a+b)2 =a2 +2ab+b2 (100+5)2 =1002 +2×100×5+52
= 10000 + 1000 + 25
= 11025
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ALGEBRA