Question

identify the terms, their coefficients for each of the following expressions,
(i) $5xy{z^2} - 3zy$
(ii) $1 + x + {x^2}$
(iii) $4{x^2}{y^2} - 4{x^2}{y^2}{z^2} + {z^2}$

Answer 1

Hint: The given expressions we have to identify the terms their coefficient each term in an algebraic expression is separated by a positive sign or negative sign for example in the term $5x$, $3y$ and $8$ when a term is made up of a constant multiplied by a variable or variable that constant is called a coefficient in the term $5x$ the coefficient is $5$ and a combination of these constant and variable connected by any or all of the four fundamental operations by positive sign negative sign multiplication by divide for example in an expression of $5x + y - 7$ here $5$ is a coefficient of $x$ and the term are $5x$ $y$ and $ - 7$ which have all different variable [or no variable] so there are no live term constant are terms without variables

Complete step-by-step answer:
STEP1: The given expressions is
(i).$5xy{z^2} - 3zy$
,(ii).$1 + x + {x^2}$
(iii)$4{x^2}{y^2} - 4{x^2}{y^2}{z^2} + {z^2}$
In the expression $5xy{z^2} - 3zy$ as we know that the terms in an expression are separated from each bother by the positive negative sign in our first expression: the two terms i.e. $5xy{z^2}$ and $3zy$
when a term is made up of a constant multiplied by a variable that constant is called a coefficient. Hence in the case of $5xy{z^2} - 3zy$ the constant multiplied in the first term is $5$ and in the second term is $ - 3$. Hence coefficients are $5$ and $ - 3$.
Step2: In case of $1 + x + {x^2}$ the terms are separated by positive sign and these terms are $1$, $x$, ${x^2}$.In this case the constant multiplied in second term is $1$ and by third term also same $1$ so coefficients are $1$ and $1$
Step3: In case of $4{x^2}{y^2} - 4{x^2}{y^2}{z^2} + {z^2}$ the one term is separated by negative sign and other term is separated by positive sign. Hence three terms are $4{x^2}{y^2}$, $4{x^2}{y^2}{z^2}$ and ${z^2}$ .constants multiplied here are $4$, $ - 4$ and $1$.So coefficients are $4$, $ - 4$ and $1$ for the terms respectively.
Step4: Hence final answer: (i) Terms are $5xy{z^2}$ and $3zy$ and coefficient is $5$ and $ - 3$
 (ii) Terms are $1$, $x$, ${x^2}$ and coefficients are $1$ and $1$
(iii) Terms are $4{x^2}{y^2}$, $4{x^2}{y^2}{z^2}$ and ${z^2}$ and coefficients are $4$, $ - 4$ and $1$

Note: In this mainly students get confused in finding coefficients and sometimes get confused between terms and expressions. Terms are the part of expression. Algebraic expression is the combination of terms separated by positive or negative sign