Algebraic Identities

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. They provide shortcuts for simplifying and manipulating algebraic expressions, saving time and reducing the risk of errors. Mastering these identities is crucial for success in algebra and related mathematical fields. They allow us to factorize expressions, solve equations more easily, and simplify complex calculations.

Formulae

• $(x+y)^2 = x^2 + 2xy + y^2$
• $(x-y)^2 = x^2 – 2xy + y^2$
• $x^2 – y^2 = (x+y)(x-y)$
• $(x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz$
• $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
• $(x-y)^3 = x^3 – 3x^2y + 3xy^2 – y^3$
• $x^3 + y^3 = (x+y)(x^2 – xy + y^2)$
• $x^3 – y^3 = (x-y)(x^2 + xy + y^2)$
• $x^3 + y^3 + z^3 – 3xyz = (x+y+z)(x^2 + y^2 + z^2 – xy – yz – zx)$

Examples

Example-1: Expand $(2a + 3b)^2$ using the appropriate identity.

Solution: Applying the identity $(x+y)^2 = x^2 + 2xy + y^2$, where $x = 2a$ and $y = 3b$, we get:

$(2a + 3b)^2 = (2a)^2 + 2(2a)(3b) + (3b)^2 = 4a^2 + 12ab + 9b^2$

Example-2: Factorize $x^2 – 9$ using the appropriate identity.

Solution: Recognizing that $x^2 – 9$ is a difference of squares ($x^2 – 3^2$), we use the identity $x^2 – y^2 = (x+y)(x-y)$. Therefore:

$x^2 – 9 = (x+3)(x-3)$

Common mistakes by students

• Incorrectly squaring terms: Students often forget to square each term inside a binomial when expanding. For example, they might write $(x+y)^2 = x^2 + y^2$ instead of $x^2 + 2xy + y^2$.
• Misinterpreting signs: Confusion with signs, particularly when dealing with $(x-y)^2$ or $(x-y)^3$.
• Forgetting the middle term: When expanding $(x+y)^2$ or $(x-y)^2$, omitting the $2xy$ term.
• Difficulty in recognizing patterns: Trouble identifying the appropriate identity to use in a given problem, especially with factorization.

Real Life Application

Algebraic identities find applications in various real-life scenarios:

• Engineering: Used in the design and construction of structures, analyzing forces, and calculations related to areas and volumes.
• Finance: Used in compound interest calculations, financial modeling, and analyzing investment returns.
• Physics: Used in solving problems related to motion, energy, and other physical phenomena.
• Computer Science: Used in algorithm optimization and in various programming tasks.

Fun Fact

The “difference of squares” identity ($x^2 – y^2 = (x+y)(x-y)$) is one of the most fundamental and frequently used identities in algebra. It can be considered as a simple form of more complex mathematical concepts.

Recommended YouTube Videos for Deeper Understanding

Q.1 Simplify $(2a + 3b)^2$.

Check Solution

Ans: B

Using the identity $(x+y)^2 = x^2 + 2xy + y^2$, where $x=2a$ and $y=3b$, we get $(2a + 3b)^2 = (2a)^2 + 2(2a)(3b) + (3b)^2 = 4a^2 + 12ab + 9b^2$.

Q.2 If $x – y = 5$ and $xy = 6$, find the value of $x^2 + y^2$.

Check Solution

Ans: C

We know $(x-y)^2 = x^2 – 2xy + y^2$. We are given $x-y=5$, so $(x-y)^2 = 5^2 = 25$. Thus, $x^2 – 2xy + y^2 = 25$. We are also given $xy=6$, so $x^2 – 2(6) + y^2 = 25$. Then $x^2 – 12 + y^2 = 25$. Therefore, $x^2 + y^2 = 25 + 12 = 37$.

Q.3 Expand $(x – 2)^3$.

Check Solution

Ans: B

Using the identity $(x-y)^3 = x^3 – 3x^2y + 3xy^2 – y^3$, where $y=2$, we get $(x-2)^3 = x^3 – 3x^2(2) + 3x(2^2) – 2^3 = x^3 – 6x^2 + 12x – 8$.

Q.4 Simplify $x^2 – 16$.

Check Solution

Ans: A

Using the identity $x^2 – y^2 = (x+y)(x-y)$, where $y=4$, we get $x^2 – 16 = x^2 – 4^2 = (x+4)(x-4)$.

Q.5 If $a+b+c=0$, find the value of $a^3 + b^3 + c^3 – 3abc$.

Check Solution

Ans: A

The identity $a^3 + b^3 + c^3 – 3abc = (a+b+c)(a^2+b^2+c^2 – ab – bc – ca)$. Since $a+b+c=0$, then $a^3 + b^3 + c^3 – 3abc = 0 \cdot (a^2+b^2+c^2 – ab – bc – ca) = 0$.

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