(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 $\frac{x}{b}$.
(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 $2x^2 - \frac{1}{2}y + 4xy$.
(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)
$\Rightarrow \text {Side} = \frac{\text{Perimeter}}{4}$
Substituting the values above, we get:
$\text {Side} = \frac{8x^2 - 2y + 16xy}{4} = 2x^2 - \frac{1}{2}y + 4xy$
(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)
$\Rightarrow \text {Breadth} = \frac{\text{Perimeter - 2(Length)}}{2}$
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 = $\frac{2mn - 4n^2}{2}$
Breadth = mn - 2n2
