I can see many questions here, and all are realtors to Algebraic Identities. So first of all, let's list out the identities which we will help us further in solving the questions.
- Identity I: (a + b)² = a² + 2ab + b²
- Identity II: (a – b)² = a² – 2ab + b²
- Identity III: a² – b²= (a + b)(a – b)
- Identity IV: (x + a)(x + b) = x² + (a + b) x + ab
- Identity V: (a + b + c)²= a² + b² + c² + 2ab + 2bc + 2ca
- Identity VI: (a + b)³ = a³ + b³ + 3ab (a + b)
- Identity VII: (a – b)³ = a³ – b³ – 3ab (a – b)
- Identity VIII: a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)
1) Identity is a true equation relating one mathematical expression to another.
In the above identity list, we can see algebraic equations are given in both sides of the equality '=' sign. Option A
━━━━━━━━━━
2) By using the algebraic identities in the above,
- x² + y² = x² + 2xy + y² (Incorrect)
- (x + y)² = x² + 2xy + y² ✓
- x³ + y³ = (x + y)(x² - xy + y²) ✓
- x² - y² = (x + y)(x - y)
So, the answer is Option A
━━━━━━━━━━
3) Yes this is identity because
⇛ x⁶ + y⁶
⇛ (x²)³ + (y²)³
Let consider a = x² and b = y²
So, by using the identity:
- a³ + b³ = (a + b)(a² - ab + b²)
⇛ (x² + y²)(x⁴ - x²y² + b⁴)
So, the correct option is Option A
━━━━━━━━━━
4) This is the part of the identity to be written in the right hand side, the left hand side is:
✏️ Refer to the above question for the same.
So, the correct option is Option C
━━━━━━━━━━
5) If we will see Identity III from the list:
- (x² - y²) = (x + y)(x - y)
And this is not the same to what is given in the Q.
So, This is not an identity because the equation is not true.
Correct option is Option C [tex].[/tex]
━━━━━━━━━━━━━━━━━━━━
