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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Practice Time 8D

1. Divide the following monomials by another monomial.

5
3
(a) 56x by 14x (b) –42xy by 7xy
4
2
2 2
3 2
2 3
(c) 54l m n by 9l m n (d) –121p q r by (–11xy z )
3 3 3
2. Divide the following binomials by the monomial.
2
(a) 24x – 6x by 3x (b) 11a b – 13ab by ab (c) 75x – 45x by 15x 5
3
6
7
3
3. Divide the following.
4
2 3
2
3
3 3
(a) 21p + 35p – 49p by 7p (b) (x y + x y – xy + xy) by xy
2
2
(c) (4x + 17x + 4) by (4x + 1) (d) 10m – 11m + 19m + 10 by (5m + 2)
3
4. Find the quotients and remainders of the following.
2
3
2
3
2
4
(a) 6a – 7a + 9a – 2a + 5 by (a + 1) (b) x + 3x – 7x – 8 by (x + x – 1)
5
4
3
6
2
2
5. Divide 6x + 5x + 4x – 3x + 2x + 2 by (x – x) by using a long division method.
Identity
An algebraic equation that is true for all values of its variable(s) is called an identity. An identity
implies that the expressions on either side of the equality sign (=) are identical.
For example, 3x + 5 = 17 is true for x = 4. If we replace x by some other number, say 2, then
3 × 2 + 5 = 11 and 11  17.
Therefore, x = 2 is not the solution of the equation 3x + 5 = 17.
2
Let us consider another equation i.e., (p + 3)(p + 2) = p + 5p + 6. p L.H.S. R.H.S.
This is true for any value of p. Such a statement containing an unknown 0 6 6
variable p that is satisfied for all values of p is called an identical equation 1 12 12
or simply identity. 2 20 20
Some Standard Identities 3 30 30
There are certain identities in mathematics which are obtained by multiplying a binomial with
another binomial. These identities are very useful in factorisation or simplification of algebraic

expressions. Some of them are as follows:
2
2
Identity 1: (a + b) = a + 2ab + b 2
i.e., Square of sum of two terms = Square of first term + 2 × (first term) × (second term)
+ Square of second term

Proof: We have,
2
(a + b) = (a + b) (a + b) = a(a + b) + b(a + b)
[Distributivity of multiplication over addition]
2
2
2
= a + ab + ba + b = a + 2ab + b 2
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