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Chapter 11: Algebraic Expressions

    Introduction

    Like and Unlike Terms

    Addition and Subtraction of Algebraic Expressions

    Multiplication of Algebraic Expressions

    Multiplication of a Monomial by a Monomial

    Multiplication of a Binomial or Trinomial by a Monomial

    Multiplication of a Binomial by a Binomial or Trinomial

    What is an Identity ?

    Some Important Identities

    Applications of Identities

    Geometric Verification of the Identities

    Exercise 11.1

    Exercise 11.2

    Exercise 11.3

    Exercise 11.4

    Exercise 11.5

    What We Have Discussed?

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Chapter 11: Algebraic Expressions > Some Important Identities

Some Important Identities

We often use algebraic identities to solve mathematical problems efficiently. These identities are also known as special products in multiplication. Let's examine three important identities involving binomial products.

Consider p+q2

p+q2 = (p + q)(p + q) = p(p + q) + q(p + q) = p2 + pq + qp + q2 = p2 + + q2

Thus p+q2 = p2 + 2pq + q2 (I)

Now, let's verify with p = 4, q = 5:

(LHS) = p+q2 = 4+52 = =

(RHS) = p2 + 2pq + q2 = 16 + 2(4)(5) + 25 = + + =

Observe that the LHS and RHS values are .

Do This

Verify the following identities using different values of m, n, r as positive integers:

(i) m−n2 = m2 - 2mn + n2

(ii) (m + n)(m - n) = m2 - n2

(iii) m+n+r2 = m2 + n2 + r2 + 2mn + 2nr + 2rm

Consider another identity: (y + z)(x + w) = xy + xz + wy + wz

(y + z)(x + w) = y(x + w) + z(x + w) = + + +

= xy + wy + xz + wz

Do This

Now take x = 3, y = 4, and w = 2, verify the identity.

• What do you observe? Is LHS = RHS?

• Try different values for x, y, w, and z to verify this identity.

• Does this identity hold true for all values of x, y, w, and z?