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. Coefficients are the numerical factors of each term.

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

• Terms are: $1$, $x$, and $x^2$.
• The coefficient of $x$ is 1.
• The coefficient of $x^2$ is 1.
• The constant term is 1.

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

• Terms are: $5xyz^2$ and $-3zy$.
• 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: $4x^2y^2$, $-4x^2y^2z^2$, and $z^2$.
• 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: $3$, $-pq$, $qr$, and $-rp$.
• The coefficient of $pq$ is -1.
• The coefficient of $qr$ is 1.
• The coefficient of $rp$ is -1.
• The constant term is 3.

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

• Terms are: $\backslash frac\{x\}\{2\}$, $\backslash frac\{y\}\{2\}$, and $-xy$.
• 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: $0.3a$, $-0.6ab$, and $0.5b$.
• 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:

• A monomial has exactly one term.
• A binomial has exactly two terms.
• A trinomial has exactly three terms.
• An expression with more than three terms is generally called a polynomial and does not fit into these specific categories.

Let's classify each expression:

• $x\; +\; y$: Contains two terms. It is a binomial.
• 1000: Contains one term. It is a monomial.
• $x\; +\; x^2\; +\; x^3\; +\; x^4$: Contains four terms. It does not fit into the categories of monomial, binomial, or trinomial. It is a polynomial.
• $7\; +\; y\; +\; 5x$: Contains three terms. It is a trinomial.
• $2y\; -\; 3y^2$: Contains two terms. It is a binomial.
• $2y\; -\; 3y^2\; +\; 4y^3$: Contains three terms. It is a trinomial.
• $5x\; -\; 4y\; +\; 3xy$: Contains three terms. It is a trinomial.
• $4z\; -\; 15z^2$: Contains two terms. It is a binomial.
• $pqr$: Contains one term. It is a monomial.
• $p^2q\; +\; pq^2$: Contains two terms. It is a binomial.
• $2p\; +\; 2q$: 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 as:

$$(ab\; -\; bc)\; +\; (bc\; -\; ca)\; +\; (ca\; -\; ab)$$

Rearranging to group like terms:

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

Combining the like terms:

$$(ab\; -\; ab)\; +\; (-bc\; +\; bc)\; +\; (-ca\; +\; ca)\; =\; 0\; +\; 0\; +\; 0\; =\; 0$$

Hence, the sum is 0.

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

We can write this as:

$$(a\; -\; b\; +\; ab)\; +\; (b\; -\; c\; +\; bc)\; +\; (c\; -\; a\; +\; ac)$$

Rearranging to group like terms:

$$a\; -\; a\; -\; b\; +\; b\; -\; c\; +\; c\; +\; ab\; +\; bc\; +\; ac$$

Combining the like terms:

$$(a\; -\; a)\; +\; (-b\; +\; b)\; +\; (-c\; +\; c)\; +\; ab\; +\; bc\; +\; ac\; =\; 0\; +\; 0\; +\; 0\; +\; ab\; +\; bc\; +\; ac$$

Hence, the sum is $ab\; +\; bc\; +\; ac$.

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

We can write this as:

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

Rearranging to group like terms:

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

Combining the like terms:

$$(2\; -\; 3)p^2q^2\; +\; (-3\; +\; 7)pq\; +\; (4\; +\; 5)\; =\; -1p^2q^2\; +\; 4pq\; +\; 9$$

Hence, the sum is $-p^2q^2\; +\; 4pq\; +\; 9$.

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

We can write this as:

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

Rearranging to group like terms:

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

Combining the like terms:

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

Hence, the sum is $2l^2\; +\; 2m^2\; +\; 2n^2\; +\; 2lm\; +\; 2mn\; +\; 2nl$.