Question

Add the following:
a. \[3mn,\, - 5mn,\,8mn,\, - 4mn\]
b. \[t - 8tz,\,3tz,\, - z,\,z - t\]
c. \[ - 7mn,\, + 5,\,12mn + 2,\,9mn - 8,\,2mn,\, - 3\]
d. \[a + b - 3,\,\,b - a + 3,\,\,a - b + 3\,\]
e. \[14x + 10y - 12xy - 13,\,\,18 - 7x - 10y + 8xy,\,\,4xy\]
f. \[5m - 7n,\,\,3n - 4m + 2,\,\,2m - 3mn - 5\]
g. \[4{x^2}y,\,\, - 3x{y^2},\,\, - 5x{y^2},\,\,5{x^2}y\]

Answer 1

Hint: Polynomials are the algebraic expressions which consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform the arithmetic operations such as addition, subtraction, multiplication and also positive integer exponents for polynomial expressions but not division by variable. An example of a polynomial with one variable is \[{x^2} + x - 12.\] In this example, there are three terms:\[{x^2},{\text{ }}x\] and \[ - 12.\] Polynomials are of 3 different types and are classified based on the number of terms in it. The three types of polynomials are which are as follow,
Monomial, Binomial and Trinomial

Monomial: A monomial is an expression which contains only one term. For an expression to be a monomial, the single term should be a non-zero term.

Binomial: A binomial is a polynomial expression which contains exactly two terms. A binomial can be considered as a sum or difference between two or more monomials.

Trinomial: A trinomial is an expression which is composed of exactly three terms.
These are the polynomial expressions. Always add the digits with same variables for example:
\[7m + 2m = 9m\]
\[7m + 2n = \]nothing.
As we know: Plus, Minus is Minus. Example: \[7 + ( - 8) = 7 - 8\].

Complete step by step solution:
(i) \[3mn + ( - 5mn) + 8mn + ( - 4mn)\]
\[ = 3mn - 5mn + 8mn - 4mn\]
\[ = - 2mn + 4mn\]
\[ = 2mn\].

(ii) \[t - 8tz + 3tz + ( - z) + (z - t)\]
\[ = t - 8tz + 3tz - z + z - t\]
\[ = t - t - 8tz + 3tz - z - z\]
\[ = 0 - 5tz + 0\]
\[ = - 5tz\].

(iii) \[ - 7mn + 5 + 12mn + 2 + 9mn - 8 + 2mn + ( - 3)\]
\[ = - 7mn + 5 + 12mn + 2 + 9mn - + 2mn - 3\]
\[ = - 7mn + 12mn + 9mn + 2mn + 5 + 2 - 8 - 3\]
\[ = 16mn - 4\].

(iv) \[(a + b - 3) + (b - a + 3) + (a - b + 3)\]
\[ = a + b - 3 + b - a + 3 + a - b + 3\]
\[ = a - a + a + b + b - b - 3 + 3 + 3\]
\[ = a + b + 3\].

(v) \[14x + 10y - 12xy - 13 + 18 - 7x - 10y + 8xy + 4xy\]
\[ = 14x - 7x + 10y - 10y - 12xy + 8xy + 4xy - 13 + 18\]
\[ = 7x + 0 - 4xy + 4xy + 5\]
\[ = 7x + 0 + 5\]
\[ = 7x + 5\].

(vi) \[5m - 7n + 3n - 4m + 2 + 2m - 3mn - 5\]
\[ = 5m - 4m + 2m - 7n + 3n + 2 - 5 - 3mn\]
\[ = 3m - 4n - 3 - 3mn\]

(vii) \[4{x^2}y - 3x{y^2} + ( - 5x{y^2}) + 5{x^2}y\]
\[ = 4{x^2}y - 3x{y^2} + 5{x^2}y - 5x{y^2}\]
\[ = 4{x^2}y + 5{x^2}y - 3{x^2} - 5x{y^2}\]
\[ = 9{x^2} - 8x{y^2}\].

Note:
 (1) We never add digits with different variables.
 (2) In the above question with which digit (coefficient) is not written, it means their coefficient value is \[1\]. For example, ‘a’, its coefficient is \[1\] i.e. \[1a\].