Ch. - 12 Algebraic Expressions
Ch. - 12 Algebraic Expressions
Exercise 12.1
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
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
(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
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
(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 y^{2} |
Coefficient of y^{2} |
(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
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
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 |
\(–7, x\) and |
Like |
Both terms have same variable \(x\) |
(iii) |
\(– 29x, – 29y\) |
\(– 29x\) and \( – 29y\) |
\(– 29, x\) and |
Unlike |
Both terms have different variables \(x\) & \(y\) |
(iv) |
\(14xy, 42yx\) |
\(14xy\) and \(42yx\) |
\(14, x,y\) and |
Like |
Both terms have same variable \(xy\) & \(xy\) |
(v) |
\(4m^2p, 4mp^2\) |
\(4m^2p\) and \(4mp^2\) |
\(4, m^2, p\) and |
Unlike |
Both terms have same variable but with different powers |
(vi) |
\(12xz, 12x^2z^2\) |
\(12xz\) and \(12x^2z^2\) |
\(12, x ,z\) and |
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
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.