Ch. - 12 Algebraic Expressions

Ch. - 12 Algebraic Expressions

Question 1

Get the algebraic expressions in the following cases using variables, constants and arithmetic operations.

(i) Subtraction of \(z\) from \(y.\)

(ii) One-half of the sum of numbers \(x\) and \(y.\)

(iii) The number \(z\) multiplied by itself.

(iv) One-fourth of the product of numbers \(p\) and \(q.\)

(v) Numbers \(x\) and \(y\) both squared and added.

(vi) Number \(5\) added to three times the product of numbers \(m\) and \(n.\)

(vii) Product of numbers \(y\) and \(z\) subtracted from \(10.\)

(viii) Sum of numbers \(a\) and \(b\) subtracted from their product.

 

Solution

Video Solution

 

Reasoning:

Let us first understand the meaning or definition of terms variable, constants and arithmetic operations

Variables are the letters used in an algebraic expression that can take any value. For e.g. \(a, b, c\) or \(z\) etc. and it can take any value which can be either \(2\) or \(5\) or any other number. Constants always have fixed values in the algebraic expressions. They cannot be assumed or changed. Arithmetic Operations are Addition, subtraction, multiplication and division.

Steps:

(i) Subtraction of \(z\) from \(y.\)

\[y - z\]

(ii) One-half of the sum of numbers \(x\) and \(y.\)

\[\frac{1}{2}\left( {x + y} \right)\]

(iii) The number \(z\) multiplied by itself.

\[z \times z = {z^2}\]

(iv) One-fourth of the product of numbers \(p\) and \(q.\)

\[\frac{1}{4}pq\]

(v) Numbers \(x\) and \(y\) both squared and added.

\[\left( {x \times x} \right) + \left( {y \times y} \right) = {x^2} + {y^2}\]

(vi) Number \(5\) added to three times the product of numbers \(m\) and \(n.\)

\[5 + 3\left( {m \times n} \right) = 5 + 3mn\]

(vii) Product of numbers \(y\) and \(z\) subtracted from \(10.\)

\[10 - \left( {y \times z} \right) = 10 - yz\]

(viii) Sum of numbers \(a\) and \(b\) subtracted from their product.

\[\left( {a \times b} \right)-\left( {a + b} \right) = ab - \left( {a + b} \right)\]

Question 2

(i) Identify the terms and their factors in the following expressions. Show the terms and factors by tree diagrams.

(a) \(x – 3\)

(b) \(1 + x + x^2\)

(c) \(y – y^3\)

(d) \(5xy^2 + 7x^2y\)

(e) \(– ab + 2b^2 – 3a^2\)

(ii) Identify terms and factors in the expressions given below:

(f) \(–4x + 5\) (b) \(–4x + 5y\) (c) \(5y + 3y^2\)

(g) \(xy + 2x^2y^2\) (e) \(pq + q\) (f) \(1.2 ab – 2.4 b + 3.6 a\)

(h) \(\frac{3}{4}x + \frac{1}{4}\) (h) \(0.1 p^2 + 0.2 q^2\)

 

Solution

Video Solution

 

(i) Identify the terms and their factors in the following expressions. Show the terms and factors by tree diagrams.

(a) \(x – 3\)

Steps:

Term \(= x\) and Factor \(= 1\)

(b) \(1 + x + x^2\)

Steps:

Term \(= x\) and Factor \(= 1;\) Term \(= x^2\) and Factor \(= 1\)

(c) \(y – y^3\)

Steps:

Term \(= y\) and Factor \(= 1;\) Term \(= y^3\) and Factor \(= –1\) 

(d) \(5xy^2 + 7x^2y\)

Steps:

Term \(= xy^2\) and Factor \(= 5;\) Term \(= x^2y\) and Factor \(= 7\)

(e) \(– ab + 2b^2 – 3a^2\)

Steps:

Term \(= ab\) and Factor \(= –1; \)Term \(= b^2\) and Factor \(= 2;\) Term \(a^2\) and Factor \(= –3\)

(ii) Identify terms and factors in the expressions given below:

(f) \(–4x + 5\) (b) \(–4x + 5y\) (c) \(5y + 3y^2\)

(g) \(xy + 2x^2y^2\) (e) \(pq + q\) (f) \(1.2 ab – 2.4 b + 3.6 a\)

(h) \(\frac{3}{4}x + \frac{1}{4}\) (h) \(0.1 p^2 + 0.2 q^2\)

S.No.

Expression

Term

Factors

a)

\(–4x + 5\)

\(-4x\) and \(5\)

\(-4, x\)and \(5\)

b)

\(-4x + 5y\)

\(-4x\) and \(5y\)

\(-4, x\) and \(5, y\)

c)

\(5y + 3y^2\)

\(5y\) and \(3y^2\)

\(5, y \) and \(3, y, y\)

d)

\(xy + 2x^2y^2\)

\(xy\) and \(2x^2y^2\)

\(x, y\) and \(2, x, x,y, y\)

e)

\(pq + q\)

\(pq\) and \(q\)

\(p, q\) and \(q\)

f)

\(1.2ab - 2.4b + 3.6a\)

\(1.2ab, -2.4b\) and \(3.6a\)

\(1.2, a, b, -2.4, b\) and \(3.6, a\)

g)

 \(\frac{{3}}{4} x + \frac{{1}}{4} \)

\(\frac{{3}}{4} x \) and \( \frac{{1}}{4}\)

\(\frac{{3}}{4}, x\) and \(\frac{{1}}{4} \)

h)

\(0.1 p^2 + 0.2 q^2\)

\(0.1p^2\) and \(0.2q^2\)

\(0.1, p, p\) and \(0.2, q, q\)

Question 3

Identify the numerical coefficients of terms (other than constants) in the following expressions:

(i) \(5 – 3t^2\)       

ii) \(1 + t + t^2 + t^3\)         

(iii) \(x + 2xy + 3y\)

(iv) \(100m + 1000n\)     

(v) \(– p^2q^2 + 7pq\)   

(vi) \(1.2a + 0.8b\)

(vii) \(3.14r^2\)         

(viii) \(2(l + b)\)                 

(ix) \(0.1y + 0.01y^2\)

 

 

Solution

Video Solution

 

 

S.No.

Expression

Term

Numerical Coefficient

(i)

\(5 - 3t^2\)

\(-3t^2\)

\(-3\)

(ii)

\(1 + t + t^2 + t^3\)

\(t, t^2 \) and \( t^3\)

\(1, 1 \) and \( 1\)

(iii)

\(x + 2xy + 3y\)

\(x , 2xy \) and \( 3y\)

\(1, 2 \) and \( 3\)

(iv)

\(100m + 1000n\)

\(100m \) and \( 1000n\)

\(100 \) and \( 1000\)

(v)

\(-p^2q^2 + 7pq\)

\(-p^2q^2 \) and \( 7pq\)

\(-1 \) and \( 7\)

(vi)

\(1.2 a + 0.8 b\)

\(1.2a \) and \( 0.8b\)

\(1.2 \) and \( 0.8\)

(vii)

\(3.14r^2\)

\(3.14r^2\)

\(3.14\)

(viii)

\(2(l + b)\)

\(2l \) and \( 2b\)

\(2 \) and \( 2\)

(ix)

\(0.1y + 0.01y^2\)

\(0.1y \) and \( 0.01 y^2\)

\(0.1 \) and \( 0.01\)

Question 4

(a) Identify terms which contain \(x\) and give the coefficient of \(x.\)

(i) \(y^2x + y\)

(ii) \(13y^2 – 8yx\)

(iii) \(x + y + 2\)

(iv) \(5 + z + zx\)

(v) \(1 + x + xy\)

(vi) \(12xy^2 + 25\)

(vii) \(7x + xy^2\)

(b) Identify terms which contain \(y^2\) and give the coefficient of \(y^2.\)

(i) \(8 – xy^2\)

(ii) \(5y^2 + 7x\)

(iii) \(2x^2y – 15xy^2\) + \(7y^2\)

 

Solution

Video Solution

 

(a) Identify terms which contain \(x\) and give the coefficient of \(x.\)

(i) \(y^2x + y\)

(ii) \(13y^2 – 8yx\)

(iii) \(x + y + 2\)

(iv) \(5 + z + zx\)

(v) \(1 + x + xy\)

(vi) \(12xy^2 + 25\)

(vii) \(7x + xy^2\)

S.No.

Expression

Term containing x

Coefficient of x

(i)

\(y^2x + y\)

\(y^2x\)

\(y^2\)

(ii)

\(13y^2 – 8yx\)

\(–8yx\)

\(–8y\)

(iii)

\(x + y + 2\)

\(x\)

\(1\)

(iv)

\(5 + z + zx\)

\(zx\)

\(z\)

(v)

\(1 + x + xy\)

\(x\) and \(xy\)

\(1\) and \(y\)

(vi)

\(12xy^2 + 25\)

\(12xy^2\)

\(12y^2\)

(vii)

\(7 x + xy^2\)

\(7 x\) and \(xy^2\)

\(7\) and \(y^2\)

 

(b) Identify terms which contain \(y^2\) and give the coefficient of \(y^2.\)

(i) \(8 – xy^2\)

(ii) \(5y^2 + 7x\)

(iii) \(2x^2y – 15xy^2\) + \(7y^2\)

S.No.

Expression

Term containing y2

Coefficient of y2

(i)

\(8 – xy^2\)

\(– xy^2\)

\(– x\)

(ii)

\(5y^2 + 7x\)

\(5y^2\)

\(5\)

(iii)

\(2x^2 y – 15xy^2 + 7y^2\)

\(– 15xy^2\) and \(7y^2\)

\(– 15x\) and \(7\)

Question 5

Classify into monomials, binomials and trinomials.

(i) \(4y – 7z\)

(ii) \(y^2\)

(iii) \(x + y – xy\)

(iv) \(100\)

(v) \(ab – a – b\)

(vi) \(5 – 3t\)

(vii) \(4p^2q – 4pq^2\)

(viii) \(7mn\)

(ix) \(z^2 – 3z + 8\)

(x) \(a^2 + b^2\)

(xi) \(z^2 + z\)

(xii) \(1 + x + x^2\)

 

Solution

Video Solution

 

Steps:

Monomial means expression having single term.

Binomials means expression having two terms.

Trinomials means expression having three terms.

S No.

Expression

No. of terms

Classification

(i)

\(4y – 7z\)

\(2\)

Binomial

(ii)

\(y^2\)

\(1\)

Monomial

(iii)

\(x + y – xy\)

\(3\)

Trinomial

(iv)

\(100\)

\(1\)

Monomial

(v)

\(ab – a – b\)

\(3\)

Trinomial

(vi)

\(5 – 3t\)

\(2\)

Binomial

(vii)

\(4p^2q – 4pq^ 2\)

\(2\)

Binomial

(viii)

\(7mn\)

\(1\)

Monomial

(ix)

\(z^2 – 3z + 8\)

\(3\)

Trinomial

(x)

\(a^2 + b^2\)

\(2\)

Binomial

(xi)

\(z^2 + z\)

\(2\)

Binomial

(xii)

\(1 + x + x^2\)

\(3\)

Trinomial

Question 6

State whether a given pair of terms is of like or unlike terms.

(i) \(1, 100\)

(ii) \(–7x, x\)

(iii) \(– 29x, – 29y\)

(iv) \(14xy, 42yx\)

(v) \(4m^2p, 4mp^2\)

(vi) \(12xz, 12x^2z^2\)

 

Solution

Video Solution
 

 

S.No.

Expression

Terms

Factors

Like/ Unlike

Reason

(i)

\(1, 100\)

\(1\) and \(100\)

\(1\) and \(100\)

Like

Bothe the terms has no variables

(ii)

\(–7x,  x\)

\(–7x\) and
\( x\)

\(–7, x\) and
\(x\)

Like

Both terms have same variable \(x\)

(iii)

\(– 29x, – 29y\)

\(– 29x\) and \( – 29y\)

\(– 29, x\) and
\(– 29, y\)

Unlike

Both terms have different variables \(x\) & \(y\)

(iv)

\(14xy, 42yx\)

\(14xy\) and \(42yx\)

\(14, x,y\) and
\(42, y,x\)

Like

Both terms have same variable \(xy\) & \(xy\)

(v)

\(4m^2p, 4mp^2\)

\(4m^2p\) and \(4mp^2\)

\(4, m^2, p\) and
\(4, m, p^2\)

Unlike

Both terms have same variable but with different powers

(vi)

\(12xz, 12x^2z^2\)

\(12xz\) and \(12x^2z^2\)

\(12, x ,z\) and
\(12,x^2 ,z^2 \)

Unlike

Both terms have same variable but with different powers

Question 7

Identify like terms in the following:

(a)

\(\begin{align}&–xy^2, –4yx^2, 8x^2, 2xy^2, 7y, –11x^2, \\&–100x, –11yx, 20x^2y, –6x^2, \\&y, 2xy, 3x\end{align}\)  

(b)

\(\begin{align}&10pq, 7p, 8q, –p^2q^2, –7qp, –100q,\\& –23, 12q^2p^2,–5p^2, 41,2405p, \\& 78qp, 13p^2q, qp^2, 701p^2 \end{align}\)

 

Solution

Video Solution

 

Reasoning:

This question is based on the concept of like terms. If there are same variable in all
the terms in the expression, then the expression has like terms. We have to ignore constants here.

Steps:

S.No.

Terms

Like terms

(i)

\(–xy^2, –4yx^2, 8x^2, 2xy^2, 7y, –11x^2, \\–100x, – 11yx, 20x^2y, –6x^2, y, 2xy, 3x\)

\(–xy^2, 2xy^2;\\ –4yx^2, 20x^2y;\\ 8x^2, –11x^2, –6x^2;\\ 7y, y;\\ –100x, 3 x;\\ –11yx, 2xy\\\)

(ii)

\(10pq, 7p, 8q, –p^2q^2, –7qp, –100q, –23, 12q^2p^2,\\ –5p^2,41, 2405p, 78qp, 13p^2q, qp^2, 701p^2\)

\(10pq, –7qp, 78qp;\\ 8q, –100q;\\ -5p^2, 701p^2;\\ 7p, 2405p;\\ –p^2q^2, 12q^2p^2;\\ -23, 41;\\ 13p^2q, qp^2\\\)

The chapter 12 begins with an introduction to Algebraic Expressions by citing some examples of algebraic expressions in one variable. Then the concept of how expressions are formed and terms of an expression are explained.Under this, the factors of a term and coefficients are discussed in detail. Then we have the discussion of like and unlike terms.Monomials, Binomials, Trinomials and Polynomials are described in the next section.Addition and subtraction of algebraic expressions is explained in the following section. Then, we have the procedure to finding the value of an expression. The last topic of the chapter is formulas and rules in mathematics written in a concise and general form using algebraic expressions.

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