## ICSE Class 8 Maths Algebraic Expressions and Identities

**Mental Maths**

Question 1.

Fill in the blanks:

(i) A symbol which has a fixed value is called a ……….

(ii) A symbol which can be given various numerical values is called ……….. or ………

(iii) The various parts of an algebraic expression separated by + or – sign are called ………..

(iv) An algebraic expression having ………… terms is called a binomial.

(v) Each of the quantity (constant or literal) multiplied together to form a product is called a ………… of the product.

(vi) The terms having same literal coefficients are called a ………… otherwise they are called …………

(vii) Degree of the polynomial is the greatest sum of the powers of ………… in each term.

(viii)An identity is an equality which is true for ………… of variables in it.

(ix) (a + b)^{2} = …………

(x) (x + a) (x + b) = x^{2} + ………… + ab.

(xi) Dividend = ………… + remainder.

Solution:

(i) A symbol which has a fixed value is called a constant.

(ii) A symbol which can be given various numerical values

is called variable or literal.

(iii) The various parts of an algebraic expression

separated by + or – sign are called terms.

(iv) An algebraic expression having two terms is called a binomial.

(v) Each of the quantity (constant or literal) multiplied together

to form a product is called a factor of the product.

(vi) The terms having same literal coefficients are called

alike terms otherwise they are called unlike terms.

(vii) Degree of the polynomial is the greatest sum of the

powers of variables in each term.

(viii)An identity is an equality which is true

for all values of variables in it.

(ix) (a + b)^{2} = a^{2} + 2ab + b^{2}.

(x) (x + a) (x + b) = x^{2} + (a + b)x + ab.

(xi) Dividend = divisor × quotient + remainder.

Question 2.

State whether the following statements are true (T) or false (F):

(i) An algebraic expression having only one term is called a monomial.

(ii) A symbol which has fixed value is called a literal.

(iii) The term of an algebraic expression having no literal factor is called its constant term,

(iv) In any term of an algebraic expression the constant part is called literal coefficient of the term.

(v) An algebraic expression is called polynomial if the powers of the variables involved in it in each term are non-negative integers.

(vi) 5x^{2}y^{2}z, 3x^{2}zy^{2}, – \(\frac{4}{5}\) zx^{2}y^{2} are unlike terms.

(vii) 5x + \(\frac{2}{x}\) + 7 is a polynomial of degree 1.

(viii) 3x^{2}y + 7x + 8y + 9 is a polynomial of degree 3.

(ix) Numerical coefficient of -7x^{3}y is 7.

(x) An equation is true for all values of variables in it.

(xi) (a – b)^{2} + 2ab = a^{2} + b^{2}.

Solution:

(i) An algebraic expression having only one term is called a monomial. True

(ii) A symbol which has fixed value is called a literal. False

Correct:

It is called constant.

(iii) The term of an algebraic expression having

no literal factor is called its constant term.

True

(iv) In any term of an algebraic expression, the constant part

is called literal coefficient of the term. False

Correct:

It is called the constant coefficient.

(v) An algebraic expression is called polynomial if the powers

of the variables involved in it in each term are non-negative integers. True

(vi) 5x^{2}y^{2}z, 3x^{2}zy^{2}, – \(\frac{4}{5}\) zx^{2}y^{2} are unlike terms.

False

Correct:

They are like terms.

(vii) 5x + \(\frac{2}{x}\) + 7 is a polynomial of degree 1. False

Correct:

∵ \(\frac{2}{x}\) has negative power of y i.e. x^{-1}.

(viii)3x^{2}y + 7x + 8y + 9 is a polynomial of degree 3. True

(ix) Numerical coefficient of -7x^{3}y is 7. False Correct:

It is -7.

(x) An equation is true for all values of variables in it. False

Correct:

It is true for some specific values of variables.

(xi) (a – b)^{2} + 2ab = a^{2} + b^{2}. True

**Multiple Choice Questions**

**Choose the correct answer from the given four options (3 to 18):**

Question 3.

The literal coefficient of -9xyz^{2} is

(a) -9

(b) xy

(c) xyz^{2}

(d) -9xy

Solution:

Literal co-efficient of -9xyz^{2} is -9 (a)

Question 4.

Which of the following algebraic expression is

(a) 3x^{2} – 2x + 7

(b) \(\frac{5 x^{3}}{2 x}\) + 3x^{2} + 8

(c) 3x + \(\frac{2}{x}\) + 7

(d) \(\sqrt{2}\)x^{2} + \(\sqrt{3}\)x + \(\sqrt{6}\)

Solution:

3x^{2} – 2x + 7 is not an algebraic expression. (c)

Question 5.

Which of the following algebraic expressions is not a monomial?

(a) 3x × y × z

(b) -5pq

(c) 8m^{2} × n ÷ 31

(d) 7x ÷ y – z

Solution:

7x ÷ y – z = \(\frac{7 x}{y}\) – z

It is not monomial.

(As it has two terms) (d)

Question 6.

Degree of the polynomial 7x^{2}yz^{2} + 6x^{3}y^{2}z^{2} – 5x + 8y is

(a) 4

(b) 5

(c) 6

(d) 7

Solution:

Degree of polynomial

7x^{2}yz^{2} + 6x^{3}y^{2}z^{2} -5x + 8y is 3 + 2 + 2 = 7 (d)

Question 7.

a(b -c) + b(c – a) + c(a – b) is equal to

(a) ab + bc + ca

(b) 0

(c) 2(ab + bc + ca)

(d) none of these

Solution:

a(b -c) + b(c – a) + c(a – b)

= ab – ac + bc – ab + ac – bc = 0 (b)

Question 8.

\(\frac{7}{5}\)xy – \(\frac{2}{3}\)xy + \(\frac{8}{9}\) xy is equal to

Solution:

\(\frac{7}{5}\)xy – \(\frac{2}{3}\)xy + \(\frac{8}{9}\) xy

= \(\frac{63 x y-30 x y+40 x y}{45}=\frac{73}{45} x y\) (a)

Question 9.

(3p^{2}qr^{3}) × (-4p^{3}q^{2}r^{2}) x (7pq^{3}r) is equal to

(a) 84p^{6}q^{6}r^{6}

(b) -84p^{6}q^{6}r^{6}

(c) 84p^{6}q^{5}r^{6}

(d) -84p^{6}q^{5}r^{6}

Solution:

(3p^{2}qr^{3}) × (-4p^{3}q^{2}r^{2}) × (7pq^{3}r)

= 3(-4) × 7p^{2+3+1} × q^{1+2+3} × r^{3+2+1}

= -84p^{6}q^{6}r^{6} (b)

Question 10.

3m × (2m^{2} – 5mn + 4n^{2}) is equal to

(a) 6m^{3} + 15m^{2}n – 12mn^{2}

(b) 6m^{3} – 15m^{2}n + 12mn^{2}

(c) 6m^{3} – 15m^{2}n – 12mn^{2}

(d) 6m^{3} + 15m^{2}n + 12mn^{2}

Solution:

3m × (2m^{2} – 5mn + 4n^{2})

= 6m^{3} – 15m^{2}n + 12mn^{2} (b)

Question 11.

Volume of a rectangular box whose adjacent edges are 3x^{2}y, 4y^{2}z and 5z^{2}x respectively is

(a) 60xyz

(b) 60x^{2}y^{2}z^{2}

(c) 60x^{3}y^{3}z^{3}

(d) none of these

Solution:

Volume = 3x^{2}y × 4y^{2}z × 5z^{2}x = 60x^{3}y^{32}z^{3} (c)

Question 12.

(x – 1) (x + 2) is equal to

(a) 2x + 3

(b) x^{2} + 2x + 2

(c) x^{2} + 3x + 2

(d) x^{2} + 2x + 3

Solution:

(x + 1) (x + 2) = x^{2} + (1 + 2)x + 1 × 2

= x^{2} + 3x + 2 (c)

Question 13.

If x + \(\frac{1}{x}\) = 2, then x^{2} + \(\frac{1}{x^{2}}\) is equal to

(a) 4

(b) 2

(c) 0

(d) none of these

Solution:

x + \(\frac{1}{x}\) = 2

Squaring both sides,

Question 14.

If x^{2} + y^{2} = 9 and xy = 8, then x + y is equal to

(a) 25

(b) 5

(c) -5

(d) ±5

Solution:

x^{2} + y^{2} = 9, xy = 8

(x + y)^{2} = x^{2} + y^{2} + 2xy

= (9)^{2} + 2 × 8

= 9 + 16 = 25 = ±(5) (d)

Question 15.

(102)^{2} – (98)^{2} is equal to

(a) 200

(b) 400

(c) 600

(d) 800

Solution:

(102)^{2} – (98)^{2}

= (102 + 98) × (102 – 98)

= 200 × 4 = 800 (d)

Question 16.

-50x^{3}y^{2}z^{2} divided by -5xyz is equal to

(a) 10xyz

(b) 10x^{2}yz

(c) -10xyz

(d) -10x^{2}yz

Solution:

\(\frac{-50 x^{3} y^{2} z^{2}}{-5 x y z}\) = 10x^{2}yz (b)

Question 17.

96 × 104 is equal to

(a) 9984

(b) 9974

(c) 9964

(d) none of these

Solution:

96 × 104 = (100 – 4) (100 + 4)

= (100)^{2} – (4)^{2}

= 10000 – 16 = 9984 (a)

Question 18.

If the area of a rectangle is 24(x^{2}yz + xy^{2}z + xyz^{2}) and its length is 8xyz, then its breadth is

(a) 3(x + y + z)

(b) 3xyz

(c) 3(x + y – z)

(d) none of these

Solution:

In a rectangle

Area = 24(x^{2}yz + xy^{2}z + xyz^{2})

Length = 8xyz

Breadth = \(\frac{24 x y z(x+y+z)}{8 x y z}\) = 3(x + y + z) (a)

**Higher Order Thinking Skills (Hots)**

Question 1.

Using the identity (a + b) = a^{2} + 2ab + b^{2}, derive the formula for (a + b + c)^{2}. Hence, find the value of (2x – 3y + 4z)^{2}.

Solution:

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a + b + c)^{2} = (a + b)^{2} + c^{2} + 2 (a + b)c

= a^{2} + b^{2} + 2ab + c^{2} + 2ca + 2bc

= a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

(2x – 3y + 4z)^{2} = (2x)^{2} + (3y)^{2} + (4z)^{2} + 2 × 2x × (-3y)

+ 2(-3y) (4z) + 2(4z) (2x)

= 4x^{2} + 9y^{2} + 16z^{2} – 12xy – 24yz + 16zx

Question 2.

Using the product of algebraic expressions, find the formulas for (a + b)^{2} and (a – b).

Solution:

(a + b)^{2} = (a + b)^{2} (a + b)

= (a^{2} + 2ab + b^{2}) (a + b)

= a(a^{2} + 2ab + b^{2}) + b(a^{2} + 2ab + b^{2})

= a^{3} + 2a^{2}b + ab^{2} + a^{2}b + 2ab^{2} + b^{3}

= a^{32} + b^{3} + 3a^{2}b + 3ab^{2}

= a^{2} + b2^{2} + 3ab(a + b)

(a – b)^{2} = (a – b) (a – b)^{2}

= (a – b) (a^{2} – 2ab + b^{2})

= a(a^{2} – 2ab + b^{2}) – b(a^{2} – 2ab + b^{2})

= a^{3} – 2a^{2}b + ab^{2} – a^{2}b + 2ab^{2} – b^{3}

= a^{3} – b^{3} – 3a^{2}b + 3ab^{2}

= a^{3} – b^{3} – 3ab(a – b)