Operations on Algebraic Expressions
95
2. (39 +7p? - 2r+4) - (4 p? - 2q +7rº-3) = ?
(a) (p? + 2q +5r+1)
(b) (11p? + q +5r+1)
(c) (-3p2-5q +9r-7)
(d) (3p+59 -9r+7)
3. (x + 5)(x-3) = ?
(a) x2 + 5x - 15 (b) x2-3x - 15 (c) x2 + 2x +15 (d) x2 + 2x - 15
4. (2x + 3)(3x - 1) = ?
(a) (6x2 + 8x - 3) (b) (6x? +7x-3) (c) 6x2 - 7x-3 (d) (6x2 - 7x + 3)
5. (x + 4)(x + 4) = ?
(a) (x² +16) (b) (x2 + 4x +16) (c) (x2 + 8x +16) (d) (x2 +16x)
6. (x-6)(x-6) = ?
(a) (x2 - 36) (b) (x2 +36)
(c) (x2 - 6x +36) (d) (x2 - 12x +36)
7. (2x + 5)(2x-5) = ?
(a) (4x2 +25) (b) (4x2 - 25) (c) (4x2 - 10x + 25) (d) (4x2 +10x -25)
8. 8a²b =(-2ab) = ?
(b) 4a²b
(c) -4ab2
(d) -4a²b
9. (2x² + 3x + 1) + (x + 1) = ?
(a) (x + 1)
(b) (2x + 1)
(c) (x + 3)
(d) (2x + 3)
10. (x² - 4x +4) + (x - 2) = ?
21
(c) (2 - x)
(d) (2 + x + x)
(a) 4 ab?
(h) v
Answers
Answer:
Algebraic Expression
• Terms are formed by the product of variables and constants, e.g.
–3xy, 2xyz, 5x2, etc.
• Terms are added to form expressions, e.g. –2xy + 5x2.
• Expressions that contain exactly one, two and three terms are
called monomials, binomials and trinomials, respectively.
• In general, any expression containing one or more terms with non-
zero coefficients (and with variables having non-negative exponents)
is called a polynomial.
• Like terms are formed from the same variables and the powers of
these variables are also the same. But coefficients of like terms
need not be the same.
• There are number of situations like finding the area of rectangle,
triangle, etc. in which we need to multiply algebraic expressions.
• Multiplication of two algebraic expressions is again an algebraic
expression.
• A monomial multiplied by a monomial always gives a monomial.
• While multiplying a polynomial by a monomial, we multiply every
term in the polynomial by the monomial using the distributive
law a ( b + c) = ab + ac.
• In the multiplication of a polynomial by a binomial (or trinomial),
we multiply term by term, i.e. every term of the polynomial is
multiplied by every term in the binomial (or trinomial) using the
distributive property.
• An identity is an equality, which is true for all values of its variables
in the equality, i.e. an identity is a universal truth.
• An equation is true only for certain values of its variables.
• Some standard identities:
(i) (a + b)
2 = a2 + 2ab + b2
(ii) (a – b)
2 = a2 – 2ab + b2
(iii) (a + b) (a – b) = a2 – b2
(iv) (x + a) (x + b) = x2 + (a + b) x + ab
(ii) Factorisation
• Representation of an algebraic expression as the product of two or
more expressions is called factorisation. Each such expression is
called a factor of the given algebraic expression.
• When we factorise an expression, we write it as a product of its
factors. These factors may be numbers, algebraic (or literal) variables
or algebraic expressions.
A formula is an equation stating a relationship between two or more
variables. For example, the number of square units in the area (A) of a
rectangle is equal to the number of units of length (l) multiplied by the
number of units of width (w). Therefore, the formula for the area of a
rectangle is A = lw.
Sometimes, you can evaluate a variable in a formula by using the given
information.
In the figure shown, the length is 9 units
and the width is 5 units.
A = lw
A = 95
A = 45
• An irreducible factor is a factor which cannot be expressed further
as a product of factors. Such a factorisation is called an irreducible
factorisation or complete factorisation.
• A factor which occurs in each term is called the common factor.
• The factorisation done by using the distributive law (property) is
called the common factor method of factorisation.
• Sometimes, many of the expressions to be factorised are of the
form or can be put in the form: a2 + 2ab + b2, a2 – 2ab + b2, a2 – b2
or x2 + (a + b) x + ab. These expressions can be easily factorised
using identities:
a2 + 2ab + b2 = (a + b)
2
a2 – 2ab + b2 = (a – b)
2
a2 – b2 = (a + b) (a – b)
x2 + (a + b) x + ab = (x + a) (x + b)
Example 1 : Which is the like term as 24a2bc?
(a) 13 × 8a × 2b × c × a (b) 8 × 3 × a × b × c
(c) 3 × 8 × a × b × c × c (d) 3 × 8 × a × b × b × c
Solution : The correct answer is (a).
Example 2 : Which of the following is an identity?
(a) (p + q)
2 = p2 + q2 (b) p2 – q2 = (p – q)
2
(c) p2 – q2 = p2 + 2pq – q2 (d) (p + q)
2 = p2 + 2pq + q2
Solution : The correct answer is (d).
In examples 7 to 9, state whether the statements are true (T) or false (F).
Example 7 : An identity is true for all values of its variables.
Solution : True.
Example 8 : Common factor of x2y and – xy2 is xy.
Solution : True.
Example 9 : (3x + 3x2) ÷ 3x = 3x2
Solution : False.
Example 10 : Simplify (i) – pqr (p2 + q2 + r2)
(ii) (px + qy) (ax – by)
Solution : (i) – pqr (p2 + q2 + r2)
= – (pqr) × p2 – (pqr) × q2 – (pqr) × r2
= – p3qr – pq3r – pqr3
(ii) (px + qy) (ax – by)
= px (ax – by) + qy (ax – by)
= apx2 – pbxy + aqxy – qby2
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