CBSE Class 8 Maths Chapter 9: Algebraic Expressions and Identities NCERT Solutions

Solution:

To identify terms and coefficients, we look at each expression separately. Terms are parts of an expression separated by '+' or '-' signs. The coefficient is the numerical factor multiplying the variable(s) in a term.

(i) For the expression $1\; +\; x\; +\; x^2$:

• Terms are: 1, $x$, $x^2$.
• The coefficient of $x$ is 1.
• The coefficient of $x^2$ is 1.
• The constant term 1 has no variable part, so its coefficient is considered 1.

(ii) For the expression $5xyz^2\; -\; 3zy$:

• Terms are: $5xyz^2$, $-3zy$.
• The coefficient of $xyz^2$ is 5.
• The coefficient of $zy$ is -3.

(iii) For the expression $4x^2y^2\; -\; 4x^2y^2z^2\; +\; z^2$:

• Terms are: $4x^2y^2$, $-4x^2y^2z^2$, $z^2$.
• The coefficient of $x^2y^2$ is 4.
• The coefficient of $x^2y^2z^2$ is -4.
• The coefficient of $z^2$ is 1.

(iv) For the expression $3-pq+qr-rp$:

• Terms are: 3, $-pq$, $qr$, $-rp$.
• The coefficient of $pq$ is -1.
• The coefficient of $qr$ is 1.
• The coefficient of $rp$ is -1.
• The constant term 3 has no variable part.

(v) For the expression $$\backslash frac\{x\}\{2\}\; +\; \backslash frac\{y\}\{2\}\; -\; xy$$:

• Terms are: $\backslash frac\{x\}\{2\}$, $\backslash frac\{y\}\{2\}$, $-xy$.
• The coefficient of $x$ is $\backslash frac\{1\}\{2\}$.
• The coefficient of $y$ is $\backslash frac\{1\}\{2\}$.
• The coefficient of $xy$ is -1.

(vi) For the expression $0.3a\; -\; 0.6ab\; +\; 0.5b$:

• Terms are: $0.3a$, $-0.6ab$, $0.5b$.
• The coefficient of $a$ is 0.3.
• The coefficient of $ab$ is -0.6.
• The coefficient of $b$ is 0.5.

Solution:

We classify the given expressions based on the number of terms they contain:

• Monomial: An expression with only one term.
• Binomial: An expression with exactly two terms.
• Trinomial: An expression with exactly three terms.
• Polynomial: An expression with one or more terms. Expressions with more than three terms are also called polynomials and do not fit the specific categories of monomial, binomial, or trinomial.

Let's classify each expression:

(i) $x\; +\; y$: Contains two terms. It is a binomial.

(ii) 1000: Contains one term. It is a monomial.

(iii) $x\; +\; x^2\; +\; x^3\; +\; x^4$: Contains four terms. It does not fit into the monomial, binomial, or trinomial categories. It is a polynomial.

(iv) $7\; +\; y\; +\; 5x$: Contains three terms. It is a trinomial.

(v) $2y\; -\; 3y^2$: Contains two terms. It is a binomial.

(vi) $2y\; -\; 3y^2\; +\; 4y^3$: Contains three terms. It is a trinomial.

(vii) $5x\; -\; 4y\; +\; 3xy$: Contains three terms. It is a trinomial.

(viii) $4z\; -\; 15z^2$: Contains two terms. It is a binomial.

(ix) $pqr$: Contains one term. It is a monomial.

(x) $p^2q\; +\; pq^2$: Contains two terms. It is a binomial.

(xi) $2p\; +\; 2q$: Contains two terms. It is a binomial.

Solution:

To add algebraic expressions, we combine like terms. Like terms are terms that have the same variables raised to the same powers.

(i) Add $ab-bc$, $bc-ca$, and $ca-ab$:

We can write this vertically, aligning like terms:

$$ab\; -\; bc$$

$$+\; bc\; -\; ca$$

$$+\; ca\; -\; ab$$

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$$0\; +\; 0\; +\; 0\; =\; 0$$

The sum is 0.

(ii) Add $a-b+ab$, $b-c+bc$, and $c-a+ac$:

Writing vertically:

$$a\; -\; b\; +\; ab$$

$$+\; b\; -\; c\; +\; bc$$

$$+\; c\; -\; a\; +\; ac$$

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Combining like terms: $(a\; -\; a)\; +\; (-b\; +\; b)\; +\; (-c\; +\; c)\; +\; ab\; +\; bc\; +\; ac$

$$=\; 0\; +\; 0\; +\; 0\; +\; ab\; +\; bc\; +\; ac$$

$$=\; ab\; +\; bc\; +\; ac$$

The sum is $ab\; +\; bc\; +\; ac$.

(iii) Add $2p^2q^2-3pq+4$ and $5+7pq-3p^2q^2$:

Aligning like terms:

$$2p^2q^2\; -\; 3pq\; +\; 4$$

$$-\; 3p^2q^2\; +\; 7pq\; +\; 5$$

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Combine the coefficients of like terms:

$$(2\; -\; 3)p^2q^2\; +\; (-3\; +\; 7)pq\; +\; (4\; +\; 5)$$

$$=\; -1p^2q^2\; +\; 4pq\; +\; 9$$

$$=\; -p^2q^2\; +\; 4pq\; +\; 9$$

The sum is $-p^2q^2\; +\; 4pq\; +\; 9$.

(iv) Add $l^2\; +\; m^2$, $m^2\; +\; n^2$, and $n^2\; +\; l^2\; +\; 2lm\; +\; 2mn\; +\; 2nl$:

Aligning like terms:

$$l^2\; +\; m^2$$

$$+\; m^2\; +\; n^2$$

$$+\; l^2\; +\; n^2\; +\; 2lm\; +\; 2mn\; +\; 2nl$$

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Combine the coefficients of like terms:

$$(l^2\; +\; l^2)\; +\; (m^2\; +\; m^2)\; +\; (n^2\; +\; n^2)\; +\; 2lm\; +\; 2mn\; +\; 2nl$$

$$=\; 2l^2\; +\; 2m^2\; +\; 2n^2\; +\; 2lm\; +\; 2mn\; +\; 2nl$$

The sum is $2l^2\; +\; 2m^2\; +\; 2n^2\; +\; 2lm\; +\; 2mn\; +\; 2nl$.