Exploring Algebraic Identities Class 9 Notes Maths Chapter 4

Students often refer to Class 9 Maths Notes and Chapter 4 Exploring Algebraic Identities Class 9 Ganita Manjari Notes during last-minute revisions.

Class 9 Maths Chapter 4 Exploring Algebraic Identities Notes

Class 9 Maths Ganita Manjari Chapter 4 Notes

Ganita Manjari Class 9 Chapter 4 Notes – Class 9 Exploring Algebraic Identities Notes

→ Identities are equations that are true for all values of the variables.

→ One of the ways to visualise identities is using geometrical models or algebra tiles.

→ Identities can also be used to factor algebraic expressions.

→ Factorisation of quadratic expressions may be visualised by means of algebra tiles.

→ Identities can also be used to simplify calculations, such as squaring numbers or evaluating products of numbers.

→ Rational algebraic expressions may be simplified by factorisation and removing the common factors in the numerator and denominator, provided such a factor exists and it is not equal to zero.

→ (x + y)² = x² + 2xy + y²

→ (x – y)² = x² – 2xy + y²

→ (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

→ (x + y)(x – y) = x² – y²

→ (x + a)(x + b) = x² + (a + b)x + ab

→ (ax + b)(cx + d) = acx² + (ad + bc)x + bd

→ x³ – y³ = (x – y)(x² + xy + y²)

→ x³ + y³ = (x + y) (x² – xy + y²)

→ (x + y)³ = x³ + 3x²y + 3xy² + y³

→ (x – y)³ = x³ – 3x²y + 3xy² – y³

→ x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – xz – yz)

→ Identities are equations that are true for all values of the variables.

→ One of the ways to visualise identities is using geometrical models or algebra tiles.

→ Identities can also be used to factor algebraic expressions.

→ Factorisation of quadratic expressions may be visualised by means of algebra tiles.

→ Identities can also be used to simplify calculations, such as squaring numbers or evaluating products of numbers.

→ Rational algebraic expressions may be simplified by factorisation and removing the common factors in the numerator and denominator, provided such a factor exists and it is not equal to zero.

Identities
Identities are equations that are true for all values of the variables. An algebraic identity is an equation that is true for all values of the variables involved, whereas an equation need not be true for all values.

Visualising Identities
There is a way to visualise identities using geometrical models or algebra tiles.
Identity: (a + b)² = a² + 2ab + b²

Observe that the area of the outer square is (a + b)² square units.
The area of the larger square inside the outer square is a² square units, while the area of the smaller square is b² square units.
The areas of the two rectangles are ab square units each.
Together, they make the bigger square; hence, we can conclude that (a + b)² = a² + 2ab + b²

Identity: (a – b)² = a² – 2ab + b²

The area of the big square is a² square units.
The small square has an area of (a – b)² square units.
The larger rectangle has the area ab square units, while the smaller rectangle has the area b(a – b) square units.
Thus, to obtain (a – b)², we can subtract the areas of the rectangles from the big square.
We obtain (a – b)² = a² – ab – b(a – b) = a² – ab – ba + b² = a² – 2ab + b²
Hence, we can conclude that (a – b)² = a² – 2ab + b²

Identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

From the arrangement of squares and rectangles, a big square of side (a + b + c) units is obtained.
The area of big square = (a + b + c)²
Also, the area of a big square = the sum of the areas of all the squares and rectangles used to make the big square
= a² + b² + c² + ab + ab + bc + bc + ca + ca
= a² + b² + c² + 2ab + 2bc + 2ca
Hence, (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Factorisation of Algebraic Expressions Using Identities
Algebraic identities are formulas that help us recognise patterns and factor or expand expressions quickly, instead of performing lengthy calculations.
Key Identities for Factorisation

• Square of a binomial: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b²
• Difference of squares: a² – b² = (a – b)(a + b)
• Sum and difference of cubes: a³ + b³ = (a + b)(a2² – ab + b²) and a³ – b³ = (a – b)(a² + ab + b²)

Simplifying Rational Expressions
A rational algebraic expression is a fraction in which both the numerator and the denominator are polynomials.
To simplify, we:
Factorise the numerator and denominator completely.
Cancel common factors (ensuring that the denominator ≠ 0).
Write the simplified form.
Example:
\(\frac{x^2-7 x+12}{5 x^2+5 x-100}=\frac{x^2-7 x+12}{5\left(x^2+x-20\right)}=\frac{(x-4)(x-3)}{5(x-4)(x+5)}\)
\(\frac{x^2-7 x+12}{5 x^2+5 x-100}=\frac{x-3}{5(x+5)}\)

Errors	Corrections
1. Incorrectly expanding binomials. For example, when expanding (a + b)2, the result is written as (a2 + b2) instead of (a2 + 2ab + b2).	The correct expansion of (a + b)2 is (a2 + 2ab + b2). Always remember to include the cross-product term (2ab).
2. Misunderstanding the difference of squares. For example, when factoring (a2 – b2), it is written as (a – b)(a – b) instead of (a – b)(a + b).	The correct factorisation of (a2 – b2) is (a – b)(a + b). Remember, the difference of squares always factors into a product of binomials with opposite signs.
3. Incorrect application of the sum of cubes. For example, using a3 + b3 = (a + b)3 instead of the correct identity a3 + b3 = (a + b)(a2 – ab + b2).	The correct factorisation for (a3 + b3) is (a + b)(a2 – ab + b2). Always factor the sum of cubes as this product.
4. Confusing the formula for (a + b + c)2 with (a2 + b2 + c2). For example, writing (a + b + c)2 = (a2 + b2 + c2) instead of the correct formula (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.	The correct expansion of (a + b + c)2 is a2 + b2 + c2 + 2ab + 2bc + 2ca. Be sure to include the mixed product terms (2ab), (2bc), and (2ca).
5. Incorrectly expanding a binomial and applying the negative sign when expanding (a – b)2. For example, writing (a – b)2 = a2 – b2 instead of (a2 – 2ab + b2).	The correct expansion of (a – b)2 is (a2 – 2ab + b2). Always remember the negative sign in the cross-product term (-2ab).
6. Misapplying the identity for the cube of binomials. For example, using (a + b)3 = a3 + b3 instead of the correct expansion (a + b)3 = a3 + b3 + 3a2b + 3ab2.	The correct expansion for (a + b)3 is (a3 + b3 + 3a2b + 3ab2). Make sure to include the 3a2b and 3ab2 in the expansion.

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