Question

Part A: Create a fourth-degree polynomial with two terms in standard form. How do you know it is in standard form? (5 points) Part B: Explain the closure property as it relates to subtraction of polynomials. Give an example

Answer 1

Standard form of a polynomial.

The standard form of a fourth degree polynomial is

$\quad\color{blue}{f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e}$

where $a,\:b,\:c,\:d,\:e$ are real or complex and $a\neq 0$.

Solution:

A.

Ans. The fourth degree polynomial with only two terms can be any of

$\quad\quad f_{1}(x)=ax^{4}+bx^{3}$

$\quad\quad f_{2}(x)=ax^{4}+cx^{2}$

$\quad\quad f_{3}(x)=ax^{4}+dx$

$\quad\quad f_{4}(x)=ax^{4}+e$

In all four cases, $a\neq 0$.

Explanation. The above polynomials $f_{i}(x)$-s are in standard form because each of them can be written in the form of $f(x)$ by taking $(c,\:d,\:e)$, $(b,\:d,\:e)$, $(b,\:c,\:e)$ and $(b,\:c,\:d)$ respectively equal to $\color{blue}{0}$.

B.

Closer property of polynomial for subtraction. If two polynomials of equal degrees with unequal leading coefficients go through subtraction, the resulting polynomial will be of same degree.

Example.

Let, $r(x)=a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}$

and $s(x)=b_{1}x^{3}+b_{2}x^{2}+b_{3}x+b_{4}$

where $a_{i},\:b_{i}$ all belong to real or complex numbers set, $a_{1}\neq 0\neq b_{1}$ and $a_{1}\neq b_{1}$.

If subtraction is defined here, then

$\quad r(x)-s(x)=(a_{1}-b_{1})x^{3}+(a_{2}-b_{2})x^{2}+(a_{3}-b_{3})x+(a_{4}-b_{4})$

Here, $r(x)-s(x)$ is also a polynomial of fourth degree since $(a_{1}-b_{1})\neq 0$.