plz anybody give me 9th class algebraic identities
Answers
What is an Algebraic Identity?
Algebraic Identity is an equality which holds true for any real value given to the variables in the given equation i.e. for any value of the variables, the left-hand side(L.H.S) will be equal to the right-hand side(R.H.S).
Eg: (a + 1)(a +2) = a2+3a+2
=> In the given equality, let’s put a = 10 in both sides.
LHS = (a + 1) (a + 2) = (10 + 1) (10 + 2) = 11 × 12 = 132
RHS = a2 + 3a + 2 = 102 + 3 × 10 + 2 = 100 + 30 + 2 = 132
For a = 10, L.H.S = R.H.S
Similarly, you can check for any value of a and you will see L.H.S is always to equal to R.H.S, which makes this equality an identity.
An equality true for every value of the variable in it is called an identity.
Is every equation which holds true is an identity?
The answer is “NO”. The reason is that an equation is true for only certain values of the variable in it. It is not true for all values of the variable.
For eg: a2 + 3a + 2 = 132.
=> It is true for a =10 but not for a = 0 or any other value. So, it is an equation but not an identity.
Now, when you know about identities, shouldn’t we enhance our knowledge by learning some common algebraic identities?
Standard Algebraic Identities:
1. (a+b)2 = a2 + 2ab + b2
2. (a-b)2 = a2 – 2ab + b2
3. (a+b) (a-b) = a2 -b2
4. (x + a)(x + b) = x2 + (a + b) x + ab
5. (x + a)(x – b) = x2 + (a – b) x – ab
6. (x – a)(x + b) = x2 + (b – a) x – ab
7. (x – a)(x – b) = x2 – (a + b)x + ab
8. (a + b)3 = a3 + b3 +3ab (a + b)
9. (a – b)3 = a3 – b3 – 3ab (a – b)
10. (x + y + z)2 = x2 + y2 + z2 + 2xy +2yz + 2xz
11. (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
12. (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
13. (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
14. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz -xz)
15. x2 + y2 = (1/2)[(x+y)2 + (x-y)2]
16. (x+a) (x+b) (x+c) = x3+ (a + b + c ) x2 + (ab + bc +ca) x + abc
17. x3 + y3 = (x + y) ( x2 – xy + y2)
18. x3 – y3 = (x – y) ( x2 + xy + y2)
19. x2 + y2 + z2 -xy -yz -zx = (1/2)[(x-y)2 + (y -z)2 + (z -x)2]
This list of standard algebraic identities appears to be intimidating but need not worry we are going to solve questions based on these identities as well which will make these much easier for you.
1. (a+b)2 = a2 + 2ab + b2
Let’s prove this identity:
(a+b)2 = ( a + b) ( a + b)
= a ( a + b) + b ( a + b) (Multiplying every term with each other)
= a2 + ab + ba + b2
= a2 + 2ab + b2
Further, let a = 1 and b= 2
(1+2)2 = 12 + 2*1*2 + 22
9 = 9
L.H.S = R.H.S
Hence Proved
2. (a – b)2 = a2 – 2ab + b2
= (a – b) (a – b) = a (a – b) – b (a – b)
= a2 – ab – ba + b2
= a2 – 2ab + b2
Let a = 1 and b = 2
(1-2)2 = 12– 2*1*2 + 22
1 = 1
L.H.S = R.H.S
Hence Proved
3. (a+b) (a-b) = a2 -b2
= (a + b) (a – b) = a (a – b) + b (a – b)
= a2 – ab + ba – b2
= a2 – b2
Let a = 1 and b = 2
(1+2)(1-2) = 12– 22
-3 = -3
L.H.S = R.H.S
Hence Proved
In a similar way, you can prove every identity and you will see that these all satisfy the condition to be an identity. But I doubt this won’t fulfill the purpose of making you learn these identities so that you can use them while solving problems.
How to remember algebraic identities?
One you can do this by learning through memorization, by rote learning which is followed by repetition. You can repeat all the identities again and again orally or by writing until you cram them effectively but I wonder you will end committing mistake most of the time by following this way.
Another way which I think can prove more effective is by practicing problems after once you go through every identity.
Now, when you know about every identity since you are following me from start. I think it’s the best time to put our hands and mind into some problems and definitely you will see, more you will practice more easily you will get used to these identities.
Sample Problems:
Example 1: Find the product of (x + 2)(x + 2) using standard algebraic identities.
Solution: (x + 2)(x + 2) can be written as (x + 2)2.
Thus, after comparing with standard identity here a = x and b = 2. So we have,
(x + 2)2 = (x)2 + 2(x)(2) + (2)2 = x2 + 4x + 4.