Algebraic Expression, Identities & Factorisation exampler

Explanation:

(b) binomial

Here, we have to decide what is the result of the multiplication of a monomial and binomial.

As we know that, monomial is one type of polynomial contains only one term and binomial is polynomial contains two terms.

So, lets take an example to understand the above statement clearly 

Let’s take monomial = 2x and binomial = x + y

Hence, the product of monomial and binomial 

= (2x) × (x + y)

= 2x2 + 2xy

Which is a binomial.

Explanation:

(b) positive integers

Here, in the above question, we have to find what is the variable in the exponential form.

We know that, binomial is one type of a polynomial which has only two terms.

And the exponential should be always positive integers otherwise the number of terms will increase or decrease.

Explanation:

(b) (a – b)2 = a2 – 2ab + b2

Here, in the above question, we have to tell which of the following option is correct from the given options.

Now, we know that (a-b)² can also be written in the below form

= (a – b) × (a – b)

= a × (a – b) – b × (a – b)

= a2 – ab – ba + b2

= a2 – 2ab + b2

Hence, the correct option is b.

Explanation:

(d) – 5pq

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that, we can only able to add those terms which have same variables.

So, in the question, both the terms have same variables as pq and both the terms are monomial because they have only one term.

We can add both the monomial,

= – 7pq + 2pq

= (-7 + 2) pq

= -5pq

Hence, the correct option is d.

Explanation:

(d) 4x2y2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that, we can only able to add those terms which have same variables.

So, in the question, both the terms have same variables as x²y² and both the terms are monomial because they have only one term.

We can subtract both the monomial,

= x2y2 – (- 3x2y2)

= x2y2 + 3x2y2

= x2y2 (1 + 3)

= 4x2y2

Hence, the correct option is d.

Explanation:

(b) – 6m3n2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that, like terms have the variable same and the power of the terms are also same. 

So, in the question, the terms have variables as m³n².

Now, from the options we can say that the option b has the same variables as m³n².

Hence, the correct option is b.

Explanation:

(d) 6 (a2 + b)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that, binomial are those polynomials which has two variable terms.

So, from the options given to us, option d has two variable terms.

= 6 (a2 + b)

= 6a2 + b

Hence, the correct option is d.

Explanation:

(a) 2c + ab – ac – bc

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that, addition of expression is possible when the variable terms are same.

So, thee first expression have variable a and b, second expression have variable b and c, third expression have variable a and c.

So, if we have to add all the three expressions, then they include all the variables.

So, addition is possible, now adding them

= (a – b + ab) + (b + c – bc) + (c – a – ac)

= a – b + ab + b + c – bc + c – a – ac

= (a – a) + (-b + b) + (c + c) + ab – bc – ac

= 2c + ab – bc – ac

Hence, the correct option is a.

Explanation:

(a) 196 p2q4

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, while doing multiplication, it is not necessary that the terms should be like terms.

Here, we can multiply all the given three terms easily,

= 4p × (– 7q3) × (–7pq)

= (4 × (-7) × (-7)) × p × q3 × pq

= 196p2q4

Hence, the correct option is a.

Explanation:

(b) 24ab3

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, while doing multiplication, it is not necessary that the terms should be like terms.

Here, we can multiply both length and breadth terms easily,

So, now applying the equation for the area of rectangle,

Area of a rectangle = length × breadth

Now, in the question length and breadth is given as 4ab and 6b2 respectively.

= 4ab × 6b2

= 24ab3

Hence, the correct option is b.

Explanation:

(a) 12a3bc2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, while doing multiplication, it is not necessary that the terms should be like terms.

Here, we can multiply all the given three terms easily,

So, now applying the formula for the volume of cuboid.

Volume of cuboid = length × breadth × height

Now, in the question, length, breadth and height is given as 2ab, 3ac, and 2ac respectively.

= 2ab × 3ac × 2ac

= (2 × 3 × 2) × ab × ac × ac

= 12a3bc2

Hence, the correct option is a.

Explanation:

(b) 12a3b – 14ab2 + 10a2b2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, while doing multiplication, it is not necessary that the terms should be like terms.

Here, we can multiply all the given terms easily,

Now, the product of trinomial and monomial,

= (6a2 – 7b + 5ab) × 2ab

= (2ab × 6a2) – (2ab × 7b) + (2ab × 5ab)

= 12a3b – 14ab2 + 10a2b2

Hence, the correct option is b.

Explanation:

(d) 9x2 + 16y2 – 24xy

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that to find the square of any term we have to multiply that term with itself.

So, the given term is 3x – 4y.

Then, the square of this term is (3x – 4y)2

Now, the formula to find the square is

= (a – b)2 

= a2 – 2ab + b2

Here, we know that, a = 3x, b = 4y

So,

(3x – 4y)2 = (3x)2 – (2 × 3x × 4y) + (4y)2

= 9x2 – 24xy + 16y2

Hence, the correct option is d.

Explanation:

(b) – 5xyz2, 7xyz2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know that, like terms have the variable same and the power of the terms are also same. 

So, in the option b, both the terms have same variables as xyz².

Hence, the correct option is d.

Explanation:

(c) -1/3

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, the numerical values with the variable terms is known as the co-efficient of thst term.

Here, the co-efficient of -y/3 is (-1/3)

Hence, the correct option is c.

Explanation:

(c) (a + b) (a – b)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, it is a formula of (a2 – b2) = (a + b) (a – b).

Hence, the correct option is c.

Explanation:

(b) 17ab

Here, in the above question, we have to tell which of the following option is correct from the

given options.

In the above question, we have to find the common factor from all the given terms.

We know that, the variable which is present in all the terms is known as the common factor.

So, we can write all the terms in factorization form

17abc = 17 × a × b × c

34ab2 = 2 × 17 × a × b × b

51a2b = 3 × 17 × a × a × b

So, the common factors in all the three terms are 

=17 × a × b

= 17ab

Hence, the correct option is b.

Explanation:

(c) 81x2 + 49x2y2 –126x2y

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know the formula to find the square of any binomial.

Here, the given expression is in the form of binomial, binomial are the polynomial which have only two terms.

So, using the formula, we can write

= (a – b)2 

= a2 – 2ab + b2

Here, in the question, the values of a and b is given as 9x and 7xy respectively.

So,

(9x – 7xy)2 = (9x)2 – (2 × 9x × 7xy) + (7xy)2

= 81x2 – 126x2y + 49x2y2

Hence, the correct option is c.

Explanation:

(a) (23x + 54) (y – 2)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, to do factorization we always try to take out common factors from all the terms.

So, make all the terms in the factored form, then take out the common factors from all the terms.

Factorised form of 23xy – 46x + 54y – 108

= 23xy – (2 × 23x) + 54y – (2 × 54)

Taking out all the common factors,

= 23x (y – 2) + 54 (y – 2)

Now, again take out all the common factor,

= (y – 2) (23x + 54)

Hence, the correct option is a.

Explanation:

(b) (r – 7) (r – 3)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, to do factorization we always try to take out common factors from all the terms.

So, make all the terms in the factored form, then take out the common factors from all the terms.

Factorised form of r2 – 10r + 21 is 

= r2 – 7r – 3r + 21

Taking out all the common factors,

= r (r – 7) – 3 (r – 7)

Now, again take out all the common factor,

= (r – 7) (r – 3)

Hence, the correct option is b.

Explanation:

(a) (p – 19) (p + 2)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, to do factorization we always try to take out common factors from all the terms.

So, make all the terms in the factored form, then take out the common factors from all the terms.

Factorised form of p2 – 17p – 38 is = p2 – 19p + 2p – 38

Taking out all the common factors,

= p (p – 19) + 2 (p – 19)

Now, again take out all the common factor,

= (p – 19) (p + 2)

Hence, the correct option is a.

Explanation:

(c) ½pr

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Here, we know that, division is possible only when the terms are like terms and one important thing is that in division the power on the terms doesn’t matters.

So, to divide the terms 57p2qr by 114pq,

First, we have to convert it into the factorization form.

So, the factorization form is (57 × p × p × q × r)/(114 × p × q)

= 57pr/114

Now, divide both, numerator and denominator by 57.

= ½pr

Hence, the correct option is c.

Explanation:

(c) p + 2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Here, we know that, division is possible only when the terms are like terms and one important thing is that in division the power on the terms doesn’t matters.

So, to divide the terms p(4p2 – 16) by 4p (p – 2)

= (p((2p)2 – (4)2))/ (4p(p – 2))

= ((2p – 4) × (2p + 4))/(4(p – 2))

Take out the common factors

= ((2(p – 2)) × (2 (p + 4)))/(4(p -2))

= (4(p – 2)(p + 2))/ (4(p – 2))

= p + 2 

Hence, the correct option is c.

Explanation:

(a) 1

Here, in the above question, we have to tell which of the following option is correct from the

given options.

In the above question, we have to find the common factor from all the given terms.

We know that, the variable which is present in all the terms is known as the common factor.

So, we can write all the terms in factorization form

3ab= 3×a×b

2cd= 2×c×d

Here, for the above expression, there is no common factor except 1.

Hence, the correct option is a.

Explanation:

(c) x

Here, in the above question, we have to tell which of the following option is correct from the

given options.

So, to find the irreducible factor we have to carry out the factorization of the above given expression.

The factor form, which cannot be expressed more in the factorization form is known as the irreducible factor of that term.

24x2y2 = 2 × 2 × 2 × 3 × x × x × y × y

Therefore, an irreducible factor is x and y.

Hence, the correct option is c.

Explanation:

(c) 2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, to find the factor of any expression we have to factorize that expression.

So, let’s factorize the above given expression,

(a + b)2 = (a + b) (a + b) 

So, the number of factors for the given expression is 2.

Hence, the correct option is c.

Explanation:

(b) 3 (x – 8)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, to find the factor of any expression we have to factorize that expression.

So, let’s factorize the above given expression,

= 3 (x – 8)

Hence, the correct option is b.

Explanation:

(b) (x + 2), (x – 2)

Here, in the above question, we have to tell which of the following option is correct from the

given options.

We know that, to find the factor of any expression we have to factorize that expression.

So, let’s factorize the above given expression,

X2 – 4 = x2 – 22

= (x + 2) (x – 2)

So, the number of factors for the given expression is 2.

Hence, the correct option is b.

Explanation:

(d) 3x

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Here, we know that, division is possible only when the terms are like terms and one important thing is that in division the power on the terms doesn’t matters.

So, to divide the terms (-27x²y) by (-9xy),

First, we have to convert it into the factorization form.

(– 27x2y) ÷ (– 9xy) = (-27 × x × x × y)/(- 9 × x × y)

= (27/9) x

Now, dividing both numerator and denominator by 3. 

= 3x

Hence, the correct option is d.

Explanation:

(b) x2 + 2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Here, we know that, division is possible only when the terms are like terms and one important thing is that in division the power on the terms doesn’t matters.

So, to divide the terms (2x²+4) by 2,

First, we have to convert it into the factorization form.

(2x2 + 4) ÷ 2 = (2x2 + 4)/2

= (2(x2 + 2))/2

= x2 + 2

Hence, the correct option is d.

Explanation:

(d) x2 +3x + 9

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Here, we know that, division is possible only when the terms are like terms and one important thing is that in division the power on the terms doesn’t matters.

So, to divide the term (3x³+9x²+27) by 3x,

First, we have to convert It into factorization form.

(3x3 +9x2 + 27x) ÷ 3x 

= (3x3 + 9x2 + 27x)/3x

Now, we can see that 3x is the common factor, so take it out,

= 3x (x2 + 3x + 9)/3x

= x2 + 3x + 9

Hence, the correct option is d.

Explanation:

(c) 2a2 + 2b2

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know the formula for both the terms (a+b)² and (a-b)².

So, applying the formulas we have,

(a + b)2 + (a – b)2 

= (a2 + b2 + 2ab) + (a2 + b2 – 2ab)

= (a2 + a2) + (b2 + b2) + (2ab – 2ab)

= 2a2 + 2b2

Hence, the correct option is c.

Explanation:

(a) 4ab

Here, in the above question, we have to tell which of the following option is correct from the

given options.

Now, we know the formula for both the terms (a+b)² and (a-b)².

So, applying the formulas we have,

(a + b)2 – (a – b)2 

= (a2 + b2 + 2ab) – (a2 + b2 – 2ab)

= a2 – a2 + b2 – b2 + 2ab + 2ab

= 4ab

Hence, the correct option is a.

Explanation:

The product of two terms with like signs is a positive term.

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, we know that, when two terms with the same sign is multiplied, the resultant answer will have the positive sign term.

= 3m × 2n

= 6mn

Explanation:

The product of two terms with unlike signs is a negative term.

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, we know that =, when two terms with the opposite sign are multiplied, the resultant will have the negative sign term.

= -3m × 2n

= – 6mn

Explanation:

a (b + c) = a × b + a × c. … [by using left distributive law]

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, after solving the equation given on the LHS, we can obtain the values of the RHS.

= ab + ac

Explanation:

(a – b) (a – b) = (a – b)2= a2 – 2ab + b2

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, from looking the expression given on the RHS, we can say that it is the expanded foerm of (a-b)².

So, we can write (a-b)² as (a-b)(a-b),

(a – b) (a – b)= a × (a – b) – b × (a – b)

= a2 – ab – ba + b2

= a2 – 2ab + b2

Explanation:

a2 – b2 = (a + b) (a – b)

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, we know that the expanded form of the term (a²-b²),

It can be written as (a+b)(a-b).

Explanation:

(a – b)2 + (2ab – 2b2) = a2 – b2

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, the above given expansion on the RHS can be obtained by adding (2ab-2b²) on the LHS.

= (a – b)2 + (2ab – 2b2)

= a2 + b2 – 2ab + 2ab – 2b2

= a2 – b2

Explanation:

(a + b)2 – 2ab = a2 + b2

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, we know the expanded form of (a+b)².

So, simply putting the expanded form of (a+b)² in the given equation on the LHS to find the values on the RHS. 

= (a + b)2 – 2ab

= a2 + 2ab + b2 – 2ab

= a2 + b2

Explanation:

(x + a) (x + b) = x2 + (a + b) x + ab

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, we know that, by simply multiplying both the terms on thee LHS we can easily get the values required on the RHS,

= (x + a) (x + b)

= x × (x + b) + a × (x + b)

= x2 + xb + xa + ab

= x2 + x (b + a) + ab

Explanation:

The product of two polynomials is a polynomial.

Here, in the above question, we have to fill the blank space with the appropriate answer.

We know that, after multiplication of polynomials we obtain only polynomial.

Explanation:

Common factor of ax2 + bx is x (ax + b)

Here, in the above question, we have to fill the blank space with the appropriate answer.

We know that, common factors are those factors which are present in both the terms.

Here, x is present in both the terms, so we can take it out.

Explanation:

Factorised form of 18mn + 10mnp is 2mn (9 + 5p)

Here, in the above question, we have to fill the blank space with the appropriate answer.

Now, we know that, to find the factorized form, we have to take out the common factors from all the terms.

So, in the above given expression, 2mn is the common factor, we can easily take it out.

= (2 × 9 × m × n) + (2 × 5 × m × n × p)

= 2mn (9 + 5p)