Question

What is the standard form of a polynomial $\left( {8x - 4{x^2} + {x^3}} \right) - \left( {8{x^2} + 4{x^3} - 7x} \right)$?

Answer 1

An expression consisting of variables, constants and exponents that involves operations excluding division and negative exponents is called a polynomial. A polynomial is in the standard form when the terms are arranged according to their degrees, where the term with the highest degree is the first term and term with second highest degree is the second term and so on. In the question we are asked the standard form of the given polynomial so, to find the standard form we will open the parentheses and then add or subtract the like term and then we will arrange them according to their degrees.

Complete step by step answer:
Given polynomial:
$\left( {8x - 4{x^2} + {x^3}} \right) - \left( {8{x^2} + 4{x^3} - 7x} \right)$
Now we will open the parentheses and change the sign accordingly. So, we get;
$= 8x - 4{x^2} + {x^3} - 8{x^2} - 4{x^3} + 7x$
Now we will add and subtract the like terms accordingly. So, we get;
$= 15x - 12{x^2} - 3{x^3}$
Now we will arrange the terms in descending order of their degrees. So, we get;
$= - 3{x^3} - 12{x^2} + 15x$
This is the standard form of the given polynomial.

Note: If in a polynomial all the non-zero terms have the same degree then the polynomial is called a homogeneous polynomial. For example,
${x^7} + 9{x^5}{y^2} + {y^7}$
In the above polynomial the degree of each term is $7$. So, the polynomial is homogeneous.