Fill in the blanks :

(i) The highest power of the …………… in a polynomial is called its degree.

(ii) The degree of the polynomial 7⁵ is …………… .

(iii) Several parts of an algebraic expression separated by + or - signs are called the …………… of the expression.

(iv) Any number is a polynomial of degree …………… .

(v) Terms having the same …………… are called like terms.

(vi) The algebraic expression for the statement ‘the number of times b is contained in x' is …………… .

(vii) 1 + x + y + xy is a polynomial having …………… terms and degree ………….. .

(viii) The length of a side of a square having perimeter 8x² - 2y + 16xy is …………… .

(ix) In a polynomial, the exponents of the variables are always …………… .

(x) If the length of a rectangle having perimeter (6 m² - 2 mn + 2 m²n - 4n²) units, is (3m² + m²n - 2 mn) units, then its breadth is equal to …………… units.

(i) The highest power of the variable in a polynomial is called its degree.

(ii) The degree of the polynomial 7⁵ is 0.

(iii) Several parts of an algebraic expression separated by + or - signs are called the terms of the expression.

(iv) Any number is a polynomial of degree 0.

(v) Terms having the same literal coefficients are called like terms.

(vi) The algebraic expression for the statement ‘the number of times b is contained in x' is $\dfrac{x}{b}$.

(vii) 1 + x + y + xy is a polynomial having 4 terms and degree 2.

(viii) The length of a side of a square having perimeter 8x² - 2y + 16xy is $2x^2 - \dfrac{1}{2}y + 4xy$.

(ix) In a polynomial, the exponents of the variables are always non-negative integers.

(x) If the length of a rectangle having perimeter (6 m² - 2 mn + 2 m²n - 4n²) units, is (3m² + m²n - 2 mn) units, then its breadth is equal to mn - 2n² units.

Explanation

(i) In algebra, the degree represents the maximum power of the variable present. For example, in x³ + x, the degree is 3.

(ii) The degree refers to the power of the variable. Since 7⁵ is just a constant with no variable, its degree is 0.

(iii) Terms are the individual building blocks of an expression. In 3x + 5y, "3x" and "5y" are the terms.

(iv) Any constant number k can be written as k.x⁰. Since the variable power is 0, the degree is 0.

(v) Like terms must have the exact same variables raised to the exact same powers, such as 5ab² and -2ab².

(vi) To find how many times one number is "contained" in another, we use division. For example, 2 is contained in 10 five times (10 ÷ 2).

(vii) Given polynomial: 1 + x + y + xy

Terms: 1, x, y, xy = 4 terms

Degree: The term xy has degree 1 + 1 = 2.

(viii)

Given:

Perimeter of square = 8x² - 2y + 16xy

Side = ?

We have the formula:

Perimeter of a square = 4 x (Side)

$\Rightarrow \text {Side} = \dfrac{\text{Perimeter}}{4}$

Substituting the values above, we get:

$\text {Side} = \dfrac{8x^2 - 2y + 16xy}{4} = 2x^2 - \dfrac{1}{2}y + 4xy$

(ix) Polynomials are defined by having non-negative integer exponents. They cannot have variables with negative powers (x^-1) or roots ($\sqrt{x}$).

(x)

Given:

Length = (3m² + m²n - 2 mn)

Perimeter of a rectangle = (6 m² - 2 mn + 2 m²n - 4n²)

Breadth = ?

We know the formula:

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

First, find 2 x Length:

2(Length) = 2(3m² + m²n - 2 mn) = 6m² + 2m²n - 4 mn

Now, find Perimeter - 2(Length):

$\begin{array}{rcccccc} 6m^2 & + & 2m^2n & - & 2mn & - & 4n^2 \\ -6m^2 & + & 2m^2n & - & 4mn & + & 0 \\ -\phantom{6m^2} & - & & + & & - \\ \hline 0 & & 0 & & 2mn & - & 4n^2 \\ \hline \end{array}$

Perimeter - 2(Length) = 2mn - 4n²

Now, we have:

Breadth = $\dfrac{2mn - 4n^2}{2}$

Breadth = mn - 2n²