Fundamental Concepts of Algebra — Exercise 12(B)

Write all the terms of the following algebraic expressions:

$2 + 3x$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{2 + 3x}$ are $\bm{2}$ and $\bm{3x}$.

Write all the terms of the following algebraic expressions:

$a + 2b - 3c$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{a + 2b - 3c}$ are $\bm{a}$, $\bm{2b}$ and $\bm{-3c}$.

Write all the terms of the following algebraic expressions:

$3 - 2x + 5x^2$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{3 - 2x + 5x^2}$ are $\bm{3}$, $\bm{-2x}$ and $\bm{5x^2}$.

Write all the terms of the following algebraic expressions:

$1 - 8xyz$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{1 - 8xyz}$ are $\bm{1}$ and $\bm{-8xyz}$.

Write all the terms of the following algebraic expressions:

$2p^2 - 3pq + q^2 - 1$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{2p^2 - 3pq + q^2 - 1}$ are $\bm{2p^2}$, $\bm{-3pq}$, $\bm{q^2}$ and $\bm{-1}$.

Write all the terms of the following algebraic expressions:

$3x^2 - 5x + \dfrac{2}{x} - \dfrac{3}{x^2}$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{3x^2 - 5x + \dfrac{2}{x} - \dfrac{3}{x^2}}$ are $\bm{3x^2}$, $\bm{-5x}$, $\bm{\dfrac{2}{x}}$ and $\bm{-\dfrac{3}{x^2}}$.

Write all the terms of the following algebraic expressions:

$-5y^3 + 2y^2 + 8y - \dfrac{3}{4}$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{-5y^3 + 2y^2 + 8y - \dfrac{3}{4}}$ are $\bm{-5y^3}$, $\bm{2y^2}$, $\bm{8y}$ and $\bm{-\dfrac{3}{4}}$.

Write all the terms of the following algebraic expressions:

$2xy + 3yz - 5zx + \dfrac{1}{9}$

Answer

The parts of an expression separated by a + or − sign are called its terms.

Hence, the terms of $\bm{2xy + 3yz - 5zx + \dfrac{1}{9}}$ are $\bm{2xy}$, $\bm{3yz}$, $\bm{-5zx}$ and $\bm{\dfrac{1}{9}}$.

Identify monomials, binomials and trinomials from the following:

$5 + x$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$5 + x$ has 2 terms, namely 5 and $x$. So, it is a binomial.

Identify monomials, binomials and trinomials from the following:

$5x$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$5x$ has 1 term only. So, it is a monomial.

Identify monomials, binomials and trinomials from the following:

$\dfrac{5}{x}$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$\dfrac{5}{x}$ has 1 term only. So, it is a monomial.

Identify monomials, binomials and trinomials from the following:

$abc$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$abc$ has 1 term only. So, it is a monomial.

Identify monomials, binomials and trinomials from the following:

$a + b + c$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$a + b + c$ has 3 terms, namely $a, b$ and $c$. So, it is a trinomial.

Identify monomials, binomials and trinomials from the following:

$-8x$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$-8x$ has 1 term only. So, it is a monomial.

Identify monomials, binomials and trinomials from the following:

$x^2 + x - \dfrac{1}{2}$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$x^2 + x - \dfrac{1}{2}$ has 3 terms, namely $x^2, x \text{ and }-\dfrac{1}{2}$. So, it is a trinomial.

Identify monomials, binomials and trinomials from the following:

$3x + \dfrac{y}{2}$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$3x + \dfrac{y}{2}$ has 2 terms, namely $3x \text{ and } \dfrac{y}{2}$. So, it is a binomial.

Identify monomials, binomials and trinomials from the following:

$3 - \dfrac{y}{2} + z$

Answer

An expression having one term is a monomial, two terms is a binomial and three terms is a trinomial.

$3 - \dfrac{y}{2} + z$ has 3 terms, namely $3, -\dfrac{y}{2} \text{ and } z$. So, it is a trinomial.

Write down the coefficient of:

$a$ in $-5ab^2$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$-5ab^2 = a \times (-5b^2)$

Hence, the coefficient of $\bm{a}$ in $\bm{-5ab^2}$ is $\bm{-5b^2}$.

Write down the coefficient of:

$p$ in $-px$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$-px = p \times (-x)$

Hence, the coefficient of $\bm{p}$ in $\bm{-px}$ is $\bm{-x}$.

Write down the coefficient of:

$x$ in $-x$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$-x = x \times (-1)$

Hence, the coefficient of $\bm{x}$ in $\bm{-x}$ is $\bm{-1}$.

Write down the coefficient of:

$x$ in $\dfrac{1}{3}xy$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$\dfrac{1}{3}xy = x \times \dfrac{1}{3}y$

Hence, the coefficient of $\bm{x}$ in $\bm{\dfrac{1}{3}xy}$ is $\bm{\dfrac{1}{3}y}$.

Write down the coefficient of:

$xz$ in $-3xyz$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$-3xyz = xz \times (-3y)$

Hence, the coefficient of $\bm{xz}$ in $\bm{-3xyz}$ is $\bm{-3y}$.

Write down the coefficient of:

$p$ in $abcp$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$abcp = p \times abc$

Hence, the coefficient of $\bm{p}$ in $\bm{abcp}$ is $\bm{abc}$.

Write down the coefficient of:

$y^2$ in $-6axy^2$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$-6axy^2 = y^2 \times (-6ax)$

Hence, the coefficient of $\bm{y^2}$ in $\bm{-6axy^2}$ is $\bm{-6ax}$.

Write down the coefficient of:

$abc$ in $abc$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$abc = abc \times 1$

Hence, the coefficient of $\bm{abc}$ in $\bm{abc}$ is $\bm{1}$.

Write down the coefficient of:

$x^2y$ in $8abx^2yz$

Answer

Any factor of a monomial is called the coefficient of the remaining factor.

$8abx^2yz = x^2y \times 8abz$

Hence, the coefficient of $\bm{x^2y}$ in $\bm{8abx^2yz}$ is $\bm{8abz}$.

Write down the numerical as well as literal coefficient of the following monomials:

$9a^2b$

Answer

Numerical coefficient is the number (along with the sign, + or −).

Literal coefficient is the variable part in the monomials.

Hence, in $\bm{9a^2b}$ the numerical coefficient is $\bm{9}$ and the literal coefficient is $\bm{a^2b}$.

Write down the numerical as well as literal coefficient of the following monomials:

$-3xyz^2$

Answer

Numerical coefficient is the number (along with the sign, + or −).

Literal coefficient is the variable part in the monomials.

Hence, in $\bm{-3xyz^2}$ the numerical coefficient is $\bm{-3}$ and the literal coefficient is $\bm{xyz^2}$.

Write down the numerical as well as literal coefficient of the following monomials:

$\dfrac{3}{5}apq$

Answer

Numerical coefficient is the number (along with the sign, + or −).

Literal coefficient is the variable part in the monomials.

Hence, in $\bm{\dfrac{3}{5}apq}$ the numerical coefficient is $\bm{\dfrac{3}{5}}$ and the literal coefficient is $\bm{apq}$.

Write down the numerical as well as literal coefficient of the following monomials:

$-\dfrac{7}{2}pqr$

Answer

Numerical coefficient is the number (along with the sign, + or −).

Literal coefficient is the variable part in the monomials.

Hence, in $\bm{-\dfrac{7}{2}pqr}$ the numerical coefficient is $\bm{-\dfrac{7}{2}}$ and the literal coefficient is $\bm{pqr}$.

Write down the numerical as well as literal coefficient of the following monomials:

$-a^3b^2c$

Answer

Numerical coefficient is the number (along with the sign, + or −).

Literal coefficient is the variable part in the monomials.

Hence, in $\bm{-a^3b^2c}$ the numerical coefficient is $\bm{-1}$ and the literal coefficient is $\bm{a^3b^2c}$.

Write down the numerical as well as literal coefficient of the following monomials:

$xy^2z^3$

Answer

Numerical coefficient is the number (along with the sign, + or −).

Literal coefficient is the variable part in the monomials.

Hence, in $\bm{xy^2z^3}$ the numerical coefficient is $\bm{1}$ and the literal coefficient is $\bm{xy^2z^3}$.

Write all the factors of:

$3a$

Answer

In a monomial, each quantity of the given product is called a factor of the monomial.

$3a = 3 \times a$

Hence, the factors are $\bm{3}, \bm{a}$ and $\bm{3a}$.

Write all the factors of:

$-x$

Answer

In a monomial, each quantity of the given product is called a factor of the monomial.

$-x = (-1) \times x$

Hence, the factors are $\bm{-1}, \bm{x}$ and $\bm{-x}$.

Write all the factors of:

$ab$

Answer

In a monomial, each quantity of the given product is called a factor of the monomial.

$ab = a \times b$

Hence, the factors are $\bm{a}, \bm{b}$ and $\bm{ab}$.

Write all the factors of:

$3pq^2$

Answer

In a monomial, each quantity of the given product is called a factor of the monomial.

$3pq^2 = 3 \times p \times q \times q$

Taking the factors one at a time, two at a time, three at a time and all four together, we get:

One at a time : $3, p, q$

Two at a time : $3p, 3q, pq, q^2$

Three at a time : $3pq, 3q^2, pq^2$

All four together : $3pq^2$

Hence, the factors are $\bm{3}, \bm{p}, \bm{q}, \bm{3p}, \bm{3q}, \bm{pq}, \bm{q^2}, \bm{3pq}, \bm{3q^2}, \bm{pq^2}$ and $\bm{3pq^2}$.

Group the like terms together:

$5x, -6y, \dfrac{3}{5}x, -y, \dfrac{5}{7}y, x, -\dfrac{1}{2}y$

Answer

Terms having the same literal factors are like terms.

The terms containing $x$ are $5x, \dfrac{3}{5}x \text{ and } x$.

The terms containing $y$ are $-6y, -y, \dfrac{5}{7}y \text{ and }-\dfrac{1}{2}y$.

Hence, the groups of like terms are $\bm{\left(5x, \dfrac{3}{5}x, x\right)\text{ and }\left(-6y, -y, \dfrac{5}{7}y, -\dfrac{1}{2}y\right)}$.

Group the like terms together:

$xy^2, -3x^2y, 5y^2x, \dfrac{2}{3}yx^2, -\dfrac{3}{5}xy^2$

Answer

Terms having the same literal factors are like terms.

The terms containing $xy^2$ are $xy^2, 5y^2x \text{ and } -\dfrac{3}{5}xy^2$.

The terms containing $x^2y$ are $-3x^2y \text{ and } \dfrac{2}{3}yx^2$.

Hence, the groups of like terms are

$\bm{\left(xy^2, 5y^2x, -\dfrac{3}{5}xy^2\right)\text{ and }\left(-3x^2y, \dfrac{2}{3}yx^2\right)}$.

Group the like terms together:

$2p^2q, -3pq^2, 8qp^2, 6p^2q^2, 7q^2p, -8q^2p^2$

Answer

Terms having the same literal factors are like terms.

The terms containing $p^2q$ are $2p^2q$ and $8qp^2$.

The terms containing $pq^2$ are $-3pq^2$ and $7q^2p$.

The terms containing $p^2q^2$ are $6p^2q^2$ and $-8q^2p^2$.

Hence, the groups of like terms are

$\bm{\left(2p^2q, 8qp^2\right), \left(-3pq^2, 7q^2p\right)\text{ and }\left(6p^2q^2, -8q^2p^2\right)}$.

State true or false:

(i) The expression $8a$ has 2 terms.

(ii) $6 + xy$ is a trinomial.

(iii) $3ab$ is a binomial.

(iv) $5xy$ and $6axy$ are unlike terms.

(v) $a^2b$ and $-ba^2$ are like terms.

(vi) $xy$ and $-yx$ are like terms.

(vii) 6 and $6p$ are like terms.

(viii) $x$ and $-x$ are unlike terms.

(ix) $-yx$ and $2yx$ are like terms.

(x) The coefficient of $y$ in $-5xy$ is −5.

Answer

(i) False. $8a$ is a single term (a monomial); 8 and $a$ are its factors, not its terms.

(ii) False. $6 + xy$ has 2 terms, so it is a binomial.

(iii) False. $3ab$ has only 1 term, so it is a monomial.

(iv) True. The literal factors $xy$ and $axy$ are different, so the terms are unlike.

(v) True. Both have the same literal factor $a^2b$, so they are like terms.

(vi) True. Since $yx = xy$, both have the same literal factor $xy$, so they are like terms.

(vii) False. 6 has no literal factor while $6p$ has the literal factor $p$, so they are unlike terms.

(viii) False. Both $x$ and $-x$ have the same literal factor $x$, so they are like terms.

(ix) True. Both have the same literal factor $yx$, so they are like terms.

(x) False. $-5xy = y \times (-5x)$, so the coefficient of $y$ is $-5x$.