First: State the degree and tell which monomial (term) you determined to be the one as the greatest degree. Second: Classify each of the 2 polynomials. If it is NOT a polynomial, explain why you think it is not a polynomial. For example you may say: Polynomial # 4. has degree of 5 and is classified as a quintic polynomial of 4 terms. We covered this in class (see below). Explain in sentences how you determined degree and "name" or why it is NOT a polynomial. 1. 7x^4 + 5x^2 + x − 9 2. 7x^6 − 4x^3 + 1/x

As the degree of the polynomial is said to be the highest degree of any of the terms. Therefore, the degree of [tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex] is 4
The expression [tex]7x^6\:-\:4x^3\:+\:1/x[/tex] is not a polynomial as the term [tex]\frac{1}{x}[/tex] is not a polynomial as the the variable [tex]x[/tex] is in the denominator.

Step-by-step explanation:

Considering the given expressions

[tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex]

[tex]7x^6\:-\:4x^3\:+\:1/x[/tex]

As we know that

A polynomial is considered to be the sum of monomials where each monomial is said to be a term.
The degree of a polynomial is the considered to be the greatest degree of its terms.
A polynomial is normally written with the term with the highest exponent of the variable first and afterwards decreasing from left to right.
The first term of a polynomial is said to be the leading coefficient.
The degree of the polynomial is said to be the highest degree of any of the terms.

Now, considering the expression

[tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex]

Here, the first term (monomial) i.e. [tex]7x^4\:[/tex] has the highest exponent of the variable. Here, the highest exponent is 4.

As the degree of the polynomial is said to be the highest degree of any of the terms.

Therefore, the degree of [tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex] is 4.

The polynomial [tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex] can also be called a quadrinomial as it has four terms.

Also considering the second expression

[tex]7x^6\:-\:4x^3\:+\:1/x[/tex]

This expression is not a polynomial.

For an expression to be a polynomial, it must must have to carry

no square roots of variables. For example, [tex]\sqrt{y}[/tex] is not a polynomial as the variable y is inside a radical.
no fractional or negative powers on the variables, For example, [tex]5x^{-3}[/tex] is not a polynomial as the x variable has a negative exponent.
and no variables in the denominators of any fractions. For example, [tex]\frac{1}{x}[/tex] is not a polynomial as the the variable x is in the denominator.

Therefore, the expression [tex]7x^6\:-\:4x^3\:+\:1/x[/tex] is not a polynomial as the term [tex]\frac{1}{x}[/tex] is not a polynomial as the the variable x is in the denominator.

Keywords: degree, monomial, polynomial, expression

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