Exercise 9.1 Algebraic Expressions and Identities- NCERT Solutions Class 8
Algebraic Expressions and Identities
Chapter 9 Ex.9.1 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) |
\(\begin{align}&0.3a\\- &0.6ab\\&0.5b\end{align}\) |
\(\begin{align}&0.3\\- &0.6\\&0.5\end{align}\) |
Chapter 9 Ex.9.1 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
\[\begin{align} x + {x^2} + {x^3} + {x^4},\\ab + bc + cd + da \end{align} \]
Chapter 9 Ex.9.1 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 \(2\)(\({{l}^{2}}+{{m}^{2}}+{{n}^{2}}+lm+mn+nl\))
Chapter 9 Ex.9.1 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)
\(\dfrac{\begin{align}18 - 3p\! - 11q \!+\! 5pq \!- 2p{q^2} \!+ \!5{p^2}q \\ - 10 - 8p +\! 7p\! - \!3pq +\! 5p{q^2} + \!4{p^2}q\\\left(+\right)\,\,\,\,\left(+\right)\,\,\,\,\left(-\right)\,\,\,\,\,\,\left(+\right)\,\,\,\,\left(-\right)\,\,\,\,\,\left(-\right)\,\,\,\,\;\end{align}}{28\!+\!5p \!- \!18q \!+\! 8pq \!- \!7p q^2\! +\! p^2 q}\)