Fundamental Concepts of Algebra — Exercise 12(C)

Which of the following algebraic expressions are polynomials? Write the degree of each such polynomial.

(i) $4x^3 - 3x^2 + 5x - 2$

(ii) $\dfrac{3}{5}x^2 - \dfrac{5}{4}x + \dfrac{2}{3}$

(iii) $\dfrac{1}{4}x^4 + 2x^3 - x + 7$

(iv) $x^3 + x^2 - x + \dfrac{1}{x} + 5$

(v) $x^6 - 1$

(vi) $x^2 + \dfrac{1}{x^2} - 1$

(vii) $z - \dfrac{1}{\text{z}} + 3$

(viii) $y^3 - 4y + 6$

(ix) $z^2 - \dfrac{1}{8}$

Answer

An algebraic expression is a polynomial only when every power of the variable is a positive integer (or zero). No term of a polynomial can be of the form $\dfrac{1}{x}, \dfrac{1}{x^2}$, etc.

The degree of a polynomial is the highest power of the variable present in it.

(i) $4x^3 - 3x^2 + 5x - 2$

All powers of $x$ are positive integers, so it is a polynomial. The highest power of $x$ is 3.

Hence, it is a polynomial of degree 3.

(ii) $\dfrac{3}{5}x^2 - \dfrac{5}{4}x + \dfrac{2}{3}$

All powers of $x$ are positive integers, so it is a polynomial. The highest power of $x$ is 2.

Hence, it is a polynomial of degree 2.

(iii) $\dfrac{1}{4}x^4 + 2x^3 - x + 7$

All powers of $x$ are positive integers, so it is a polynomial. The highest power of $x$ is 4.

Hence, it is a polynomial of degree 4.

(iv) $x^3 + x^2 - x + \dfrac{1}{x} + 5$

The term $\dfrac{1}{x}$ does not have a positive integral power of $x$.

Hence, it is not a polynomial.

(v) $x^6 - 1$

All powers of $x$ are positive integers, so it is a polynomial. The highest power of $x$ is 6.

Hence, it is a polynomial of degree 6.

(vi) $x^2 + \dfrac{1}{x^2} - 1$

The term $\dfrac{1}{x^2}$ does not have a positive integral power of $x$.

Hence, it is not a polynomial.

(vii) $z - \dfrac{1}{z} + 3$

The term $\dfrac{1}{z}$ does not have a positive integral power of $z$.

Hence, it is not a polynomial.

(viii) $y^3 - 4y + 6$

All powers of $y$ are positive integers, so it is a polynomial. The highest power of $y$ is 3.

Hence, it is a polynomial of degree 3.

(ix) $z^2 - \dfrac{1}{8}$

All powers of $z$ are positive integers, so it is a polynomial. The highest power of $z$ is 2 and $\dfrac{1}{8}$ is a constant term.

Hence, it is a polynomial of degree 2.

Write down the degree of the following polynomials:

$2 + x + 3x^2$

Answer

The highest power of the variable is the degree of the polynomial.

In $2 + x + 3x^2$, the highest power of $x$ is 2.

Hence, the degree is 2.

Write down the degree of the following polynomials:

$x - 8$

Answer

The highest power of the variable is the degree of the polynomial.

In $x - 8$, the highest power of $x$ is 1.

Hence, the degree is 1.

Write down the degree of the following polynomials:

$6 - 5x + 2x^3$

Answer

The highest power of the variable is the degree of the polynomial.

In $6 - 5x + 2x^3$, the highest power of $x$ is 3.

Hence, the degree is 3.

Write down the degree of the following polynomials:

$4 + y^2$

Answer

The highest power of the variable is the degree of the polynomial.

In $4 + y^2$, the highest power of $y$ is 2.

Hence, the degree is 2.

Write down the degree of the following polynomials:

$y - y^3$

Answer

The highest power of the variable is the degree of the polynomial.

In $y - y^3$, the highest power of $y$ is 3.

Hence, the degree is 3.

Write down the degree of the following polynomials:

$z - z^2 + 3z^5$

Answer

The highest power of the variable is the degree of the polynomial.

In $z - z^2 + 3z^5$, the highest power of $z$ is 5.

Hence, the degree is 5.

Write down the degree of the following polynomials:

$m^3 - 8$

Answer

The highest power of the variable is the degree of the polynomial.

In $m^3 - 8$, the highest power of $m$ is 3.

Hence, the degree is 3.

Write down the degree of the following polynomials:

$3 + 2p + p^2 - 6p^4$

Answer

The highest power of the variable is the degree of the polynomial.

In $3 + 2p + p^2 - 6p^4$, the highest power of $p$ is 4.

Hence, the degree is 4.

Write down the degree of the following polynomials:

$1 - 2t + t^2 - 3t^3 + t^4$

Answer

The highest power of the variable is the degree of the polynomial.

In $1 - 2t + t^2 - 3t^3 + t^4$, the highest power of $t$ is 4.

Hence, the degree is 4.

Which of the following are polynomials? Write the degree of each:

(i) $m^2 + n^2 - mn$

(ii) $x^2 + y^2 + z^2 - xy - yz - zx$

(iii) $y^2 + z^2 - \dfrac{2}{y} + 3z$

(iv) $x + \dfrac{1}{y} + \dfrac{2}{z}$

(v) $a^2b + ab^2 - 3ab$

(vi) $x^2 + \dfrac{1}{x^2} + 4$

(vii) $\dfrac{1}{2}p^2q^2 + \dfrac{3}{4}q^2r^2 + \dfrac{5}{6}p^2r^2$

(viii) $x + \dfrac{1}{x}$

Answer

In a polynomial of two or more variables, every variable must have only positive integral powers. The degree is the highest sum of the powers of the variables in a term.

(i) $m^2 + n^2 - mn$

Degrees of the terms are 2, 2 and (1 + 1) = 2.

Hence, it is a polynomial of degree 2.

(ii) $x^2 + y^2 + z^2 - xy - yz - zx$

Degrees of the terms are 2, 2, 2, 2, 2 and 2.

Hence, it is a polynomial of degree 2.

(iii) $y^2 + z^2 - \dfrac{2}{y} + 3z$

The term $\dfrac{2}{y}$ does not have a positive integral power of $y$.

Hence, it is not a polynomial.

(iv) $x + \dfrac{1}{y} + \dfrac{2}{z}$

The terms $\dfrac{1}{y} \text{ and } \dfrac{2}{z}$ do not have positive integral powers.

Hence, it is not a polynomial.

(v) $a^2b + ab^2 - 3ab$

Degrees of the terms are (2 + 1), (1 + 2) and (1 + 1), i.e., 3, 3 and 2.

Hence, it is a polynomial of degree 3.

(vi) $x^2 + \dfrac{1}{x^2} + 4$

The term $\dfrac{1}{x^2}$ does not have a positive integral power of $x$.

Hence, it is not a polynomial.

(vii) $\dfrac{1}{2}p^2q^2 + \dfrac{3}{4}q^2r^2 + \dfrac{5}{6}p^2r^2$

Degrees of the terms are (2 + 2), (2 + 2) and (2 + 2), i.e., 4, 4 and 4.

Hence, it is a polynomial of degree 4.

(viii) $x + \dfrac{1}{x}$

The term $\dfrac{1}{x}$ does not have a positive integral power of $x$.

Hence, it is not a polynomial.

Write down the degree of the following polynomials:

$x + 2y - 3z$

Answer

In $x + 2y - 3z$, degrees of the terms are 1, 1 and 1.

Hence, the degree is 1.

Write down the degree of the following polynomials:

$xy + yz + zx$

Answer

In $xy + yz + zx$, degrees of the terms are (1 + 1), (1 + 1) and (1 + 1), i.e., 2, 2 and 2.

Hence, the degree is 2.

Write down the degree of the following polynomials:

$x^2 + xyz$

Answer

In $x^2 + xyz$, degrees of the terms are 2 and (1 + 1 + 1) = 3.

Hence, the degree is 3.

Write down the degree of the following polynomials:

$m^2n + mn^2 + 5mn - 6$

Answer

In $m^2n + mn^2 + 5mn - 6$, degrees of the terms are (2 + 1), (1 + 2), (1 + 1) and 0, i.e., 3, 3, 2 and 0.

Hence, the degree is 3.

Write down the degree of the following polynomials:

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

Answer

In $p^2q^2 + 2pq^2 - p^2q + 1$, degrees of the terms are (2 + 2), (1 + 2), (2 + 1) and 0, i.e., 4, 3, 3 and 0.

Hence, the degree is 4.

Write down the degree of the following polynomials:

$m^2n^3 + mn^2 + n^4$

Answer

In $m^2n^3 + mn^2 + n^4$, degrees of the terms are (2 + 3), (1 + 2) and 4, i.e., 5, 3 and 4.

Hence, the degree is 5.

State true or false:

(i) $2 - x^3$ is a polynomial of degree 3.

(ii) $\dfrac{2}{3} + \dfrac{3}{5}x^2 + x^3$ is not a polynomial.

(iii) $x + \dfrac{1}{x} + 3$ is a polynomial.

(iv) $xy - yz + zx$ is a polynomial in $x, y, z$ of degree 2.

(v) $m^2n + mn^2 + 2mn - 3n$ is a polynomial in $m, n$ of degree 3.

(vi) $mn - \dfrac{m}{n} + 5$ is a polynomial in $m$ and $n$.

Answer

(i) True. In $2 - x^3$ the highest power of $x$ is 3, so its degree is 3.

(ii) False. All the powers of $x$ are positive integers, so $\dfrac{2}{3} + \dfrac{3}{5}x^2 + x^3$ is a polynomial of degree 3.

(iii) False. The term $\dfrac{1}{x}$ does not have a positive integral power of $x$, so it is not a polynomial.

(iv) True. The degrees of the terms are 2, 2 and 2, so the degree is 2.

(v) True. The degrees of the terms are 3, 3, 2 and 1, so the degree is 3.

(vi) False. The term $\dfrac{m}{n}$ does not have a positive integral power of $n$, so it is not a polynomial.