Question

Identify the terms their coefficients for each of the following expressions:
A) $5xy{z^2} - 3zy$
B) $1 + x + {x^2}$
C) $4{x^2}{y^2} + ( - 4{x^2}{y^2}{z^2}) + {z^2}$

Answer 1

Hint: In this question we have identified the terms and their coefficients for each given expression. For that, first have to know about what is coefficient?
A coefficient is a number that is always multiplied by a variable and the coefficient is a real number whereas the term is regarded as complete mathematical expression which includes coefficients and variables.
By using the definition of coefficient we are going to identify the coefficient of the given expressions.

Complete step-by-step answer:
It is given in the question that $5xy{z^2} - 3zy$ which we can write as $5xy{z^2} + ( - 3zy)$.
We know that coefficient is a real number which always remains multiplied by a one or more variable.
Hence, we can say that the terms in this question are $5xy{z^2}$ and $ - 3zy$ whereas the coefficients are $5$ and $ - 3$.
$\therefore $ The coefficients for the expression $5xy{z^2} - 3zy$ is $5$ and $ - 3$.
It is given in the question $1 + x + {x^2}$ which can be written as $1 + 1x + 1{x^2}$
Since the coefficient is always a real number which usually remains multiplied by variables and the terms can be regarded as the whole expression which includes both coefficients and variables.
In the given question $1 + x + {x^2}$ the terms are, $1$, $x$ and ${x^2}$ on the other hand the coefficients are $1$, $1$, and $1$.
$\therefore $The coefficients for the expression $1 + x + {x^2}$ are $1$, $1$, and $1$.
It is given in the question that $4{x^2}{y^2} + ( - 4{x^2}{y^2}{z^2}) + {z^2}$,
The given expression can write as, $ = 4{x^2}{y^2} + ( - 4{x^2}{y^2}{z^2}) + {z^2}$
We know that coefficient is a real number which always remains multiplied by a one or more variable.
Hence, we can say that the terms in this question are $4{x^2}{y^2}$, $ - 4{x^2}{y^2}{z^2}$ and ${z^2}$ whereas the coefficients are $4$, $ - 4$ and $1$.
$\therefore $The coefficients for the expression $4{x^2}{y^2} + ( - 4{x^2}{y^2}{z^2}) + {z^2}$ are $4$, $ - 4$ and $1$.

Note: Coefficients can be regarded as those real numbers which are usually multiplied by variables in any case and it can be both positive and negative, while on the other hand terms include both numbers and variables together. For example $3$ is a coefficient and $3xy$ is a term.