Question

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Answer 1

Step-by-step explanation:

Considering the given expressions

• $7x^4\:+\:5x^2\:+\:x\:-\:9$

• $7x^6\:-\:4x^3\:+\:1/x$

Here are some facts about polynomial.

A polynomial  is said to be the sum of monomials where each monomial is considered to be a term.

We normally write a polynomial with the term having the highest exponent of the variable first and subsequently decreasing from left to right.

The degree of the polynomial is considered to be the highest degree of any of the terms.

Analyzing the first polynomial

Consider the polynomial

$7x^4\:+\:5x^2\:+\:x\:-\:9$

• Observe that the first term or a monomial i.e. 7x⁴ contains the greatest exponent of a variable x with a power of 4. Therefore, the degree of $7x^4\:+\:5x^2\:+\:x\:-\:9$ would be 4.

• Also, the polynomial has 4 terms, so it can also be called as 'quintic'.

Thus, $7x^4\:+\:5x^2\:+\:x\:-\:9$ has

• degree = 4
• Total number of terms = 4
• Polynomial of 4 terms
• also named as 'quintic' as its degree is 4

Analyzing the second polynomial

Consider the polynomial

$7x^6\:-\:4x^3\:+\:1/x$

This algebraic expression would not be considered as polynomial. For any algebraic expression to be a polynomial, following are the important factors that need to be fulfilled.

1. The polynomial must not contain square root of variables. For instance, any expression with the terms $\sqrt{v}$ would not a polynomial as the variable $v$ is inside the radical symbol.
2. The polynomial must not contain fractional or negative powers on the variables. For instance, any expression that may contain $9v^{-4}$ as one of its terms can not a polynomial as the $v$ variable has a negative exponent.
3. The polynomial must not contain any variables in the denominator of any fraction. For instance, if the expression contains $\frac{2}{v}$ in any one of its terms can not be a polynomial as the the variable  $v$  is in the denominator.

Thus, the algebraic expression $7x^6\:-\:4x^3\:+\:1/x$  would not be treated as a polynomial as the term $\frac{1}{x}$ has the variable x which is in the denominator.

Keywords: degree, polynomial, terms

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