Algebraic Expressions and Identities - NCERT Class 8 Maths
Algebraic Expressions and Identities
Question 1
Identify the terms, their coefficients for each of the following expressions.
(i) \(\quad 5xy{z^2} - 3zy\)
(ii) \(\quad 1 + x + {x^2}\)
(iii) \(\quad 4{x^2}{y^2} - 4{x^2}{y^2}{z^2} + {z^2}\)
(iv) \(\quad 3 - pq + qr - rp\)
(v) \(\begin{align}\quad \frac{x}{2} + \frac{y}{2} - xy\end{align}\)
(vi) \(\quad 0.3a - 0.6ab + 0.5b\)
Solution
What is known?
Expressions
What is unknown?
Terms and their coefficients
Reasoning:
The numerical factor of a term is called its numerical coefficient or simply coefficient.
Steps:
The terms and the respective coefficients of the given expressions are as follows.
- |
Terms |
Coefficients |
(i) |
\[\begin{align}&{{5}}xy{z^{\rm{2}}}\\&{{ -3}}zy\end{align}\] | \(\begin{align}5\\- 3\end{align}\) |
(ii) |
\(\begin{align}&1\\&x\\&{x^2}\end{align}\) | \(\begin{align}&1\\&1\\&1\end{align}\) |
(iii) |
\(\begin{align}&4{x^2}{y^2}\\- &4{x^2}{y^2}{z^2}\\&{z^2}\end{align}\) | \(\begin{align}&4\\-& 4\\&1\end{align}\) |
(iv) |
\(\begin{align}&{3}\\ - &pq\\&qr\\ - &rp\end{align}\) | \(\begin{align}&3\\- &1\\&1\\- &1\end{align}\) |
(v) |
\(\begin{align}&\frac{x}{2}\\&\frac{y}{2}\\- &xy\end{align}\) | \(\begin{align}&\frac{1}{2}\\&\frac{1}{2}\\- &1\end{align}\) |
(vi) |
\(\begin{align}&0.3a\\- &0.6ab\\&0.5b\end{align}\) |
\(\begin{align}&0.3\\- &0.6\\&0.5\end{align}\) |
Question 2
Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?
\(\begin{align}&x + y,\\ &1000,\\ & x + {x^2} + {x^3} + {x^4},\\ & 7 + y + 5x,\\ & 2y - 3{y^2},\\ & 2y - 3{y^2} + 4{y^3},\\ &5x - 4y + 3xy,\\ & 4z - 15{z^2},\\ & ab + bc + cd + da,\\ & pqr,\\ & {p^2}q + p{q^2},\\ & 2p + 2q\end{align}\)
Solution
What is known?
Expression
What is unknown?
The degree of the expression.
Reasoning:
1. Expression that contains only one term is called a monomial.
2. Expression that contains two terms is called a binomial.
3. Expression containing three terms is a trinomial and so on.
4. An expression containing, one or more terms with non-zero coefficient (with variables having non-negative integers as exponents) is called a polynomial.
5. A polynomial may contain any number of terms, one or more than one.
Steps:
The given expressions are classified as
Monomials: \(1000,\; pqr\)
Binomials:
\[\begin{align} &x + y,\;2y - 3{y^2},\\ & 4z - 15{z^2},\\ & {p^2}q + p{q^2},\\ & 2p + 2q \end{align}\]
Trinomials:
\[\begin{align} &7 + y + 5x,\\ & 2y - 3{y^2} + 4{y^3},\\ & 5x - 4y + 3xy \end{align} \]
Polynomials that do not fit in any of these categories are
\[\begin{align} x + {x^2} + {x^3} + {x^4},\\ab + bc + cd + da \end{align} \]
Question 3
Add the following.
(i)\(\quad ab\text{ }-\text{ }bc,\; bc\text{ }-\text{ }ca,\; ca\text{ }-\text{ }ab\)
(ii)\(\begin{align} \begin{Bmatrix} a - b + ab,\\ b- c + bc ,\\ c - a + ac \end{Bmatrix} \end{align}\)
(iii)\(\begin{align} \begin{Bmatrix} \quad 2{{p}^{2}}{{q}^{2}}-3pq+4,\\ 5+7pq -3{{p}^{2}}{{q}^{2}} \end{Bmatrix} \end{align}\)
(iv)\(\begin{align} \begin{Bmatrix} {l^2} + {m^2},\\ {m^2} + {n^2},{n^2} + {l^2},\\ 2lm + 2mn + 2nl \end{Bmatrix} \end{align}\)
Solution
What is known?
Expressions
What is unknown?
Addition
Reasoning:
Addition will take place between like terms.
Steps:
The given expressions written in separate rows, with like terms one below the other
and then the addition of these expressions are as follows.
(i)
\[\frac{\begin{align}\,\,&\,\,\,\,\,ab - bc\\&+\qquad bc - ca\\&+ \, - ab\; + ca\end{align}}{0}\]
Thus, the sum of the given expressions is \(0.\)
(ii)
\[\frac{\begin{align} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a-b+ab \\& +\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b\,\,\,\,\,\,\,\,\,\,\,\,\,\,-c+bc \\& +\,\,\,\,\,\,\,\,-a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+c\,\,\,\,\,\,\,\,\,\,\,\,+ac \\\end{align}}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,ab+\,\,\,\,\,\,\,\,bc\,\,\,\,\,\,\,\,+ac}\]
Thus, the sum of the given expressions is \(ab + bc + ac.\)
(iii)
\[\frac{\begin{align}2{p^2}{q^2} \;-\; 3pq\; + \;4\\+ \,\,\, - 3{p^2}{q^2}\;+\; 7pq \;+\; 5\end{align}}{{\,\,\,\,\,\, - {p^2}{q^2}\;\; + \;\;4pq\;\; +\;\; 9}}\]
Thus, the sum of the given expressions is \( - {p^2}{q^2} + 4pq + 9.\)
(iv)
\[\frac{\begin{align}& \,\,\,\,\,\,{{l}^{2}}+{{m}^{2}} \\& +\,\,\,\,+{{m}^{2}}+{{n}^{2}} \\& +{{l}^{2}}\qquad \; \;+{{n}^{2}} \\& + \qquad \qquad \quad \; 2lm+2mn+2nl \\\end{align}}{\begin{align} 2{{l}^{2}}& +2{{m}^{2}} +2{{n}^{2}} \\ &+2lm+2mn+2nl \end{align}}\]
Thus, the sum of the given expressions is \(2\)(\({{l}^{2}}+{{m}^{2}}+{{n}^{2}}+lm+mn+nl\))
Question 4
(a) Subtract \(4a - 7ab + 3b + 12\) from
\(12a - 9ab + 5b - 3\)
(b) Subtract \(3xy + 5yz - 7zx\) from
\(5xy - 2yz - 2zx + 10xyz\)
(c) Subtract \(\begin{Bmatrix}4{p^2}q - 3pq + 5p{q^2} \\ - 8p + 7q - 10\end{Bmatrix}\) from
\(\begin{Bmatrix}18 - 3p - 11q +\\ 5pq - 2p{q^2} + 5{p^2}q\end{Bmatrix}\)
Solution
What is known?
Expressions
What is unknown?
Subtraction
Reasoning:
Subtraction will take place between like terms.
Steps:
The given expressions in separate rows, with like terms one below the other and
then the subtraction of these expressions is as follows.
(a)
\(\dfrac{\begin{align}&12a\,\,\,\,\,\, - 9ab\,\,\,\,\,\, + 5b\,\,\,\,\,\, - 3\\&4a\,\,\,\,\,\, - 7ab\,\,\,\,\,\, + 3b\,\,\,\,\,\,\, + 12\\&\left(-\right)\,\,\,\,\,\,\,\,\left(+\right)\,\,\,\,\,\,\,\,\,\,\,\left(-\right)\,\,\,\,\,\,\,\,\,\,\left(-\right)\end{align}}{{\!\!\!\!\!\!\!\!8a - 2ab\,\,\,\, + 2b - 15}}\)
(b)
\(\dfrac{\begin{align}&5xy - 2yz - 2zx + 10xyz\\&3xy + 5yz - 7zx\\&\left(-\right)\,\,\,\,\,\,\,\,\left(-\right)\,\,\,\,\,\,\left(+\right)\end{align}}{{\,\,\,\,\,\,\,2xy - 7yz + 5zx + 10xyz}}\)
(c)
\(\dfrac{\begin{align}18 - 3p\! - 11q \!+\! 5pq \!- 2p{q^2} \!+ \!5{p^2}q \\ - 10 - 8p +\! 7p\! - \!3pq +\! 5p{q^2} + \!4{p^2}q\\\left(+\right)\,\,\,\,\left(+\right)\,\,\,\,\left(-\right)\,\,\,\,\,\,\left(+\right)\,\,\,\,\left(-\right)\,\,\,\,\,\left(-\right)\,\,\,\,\;\end{align}}{28\!+\!5p \!- \!18q \!+\! 8pq \!- \!7p q^2\! +\! p^2 q}\)
The chapter 9 begins with an introduction to expressions and representation of expression on a number line.Related terms such as Factors, Coefficients, Monomials, Binomials and Polynomials, Like and Unlike Terms are briefly explained.Then the addition and subtraction of algebraic expressions are discussed in detail.After this the multiplication of algebraic expressions is introduced and various cases under it such as multiplying a monomial by a monomial , by a Polynomial etc is dealt separately.The last section of the chapter deals with standard identities and their application to algebraic expressions.
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