To classify each polynomial based on its degree and number of terms, we will examine each polynomial one by one and match them to the appropriate descriptions:
1. Polynomial: [tex]\( x^5 + 5x^3 - 2x^2 + 3x \)[/tex]
- Degree: The highest exponent is 5.
- Number of Terms: There are four terms in this polynomial.
- So, the classifications are: "quintic" and "four terms".
2. Polynomial: [tex]\( 5 - t - 2t^4 \)[/tex]
- Degree: The highest exponent is 4.
- Number of Terms: There are three terms in this polynomial.
- So, the classifications are: "quartic" and "trinomial".
3. Polynomial: [tex]\( 8y - \frac{6y^2}{7^3} \)[/tex]
- Degree: The highest exponent is 2.
- Number of Terms: There are two terms in this polynomial.
- So, the classifications are: "quadratic" and "binomial".
4. Polynomial: [tex]\( 2x^5 y^3 + 3 \)[/tex]
- Degree: The sum of the exponents in the first term is 5 + 3 = 8.
- Number of Terms: There are two terms in this polynomial.
- So, the classifications are: "8th degree" and "binomial".
5. Polynomial: [tex]\( 4m - m^2 + 1 \)[/tex]
- Degree: The highest exponent is 2.
- Number of Terms: There are three terms in this polynomial.
- So, the classifications are: "quadratic" and "trinomial".
6. Polynomial: [tex]\( -2g^2h \)[/tex]
- Degree: The sum of the exponents in the term is 2 + 1 = 3.
- Number of Terms: There is only one term in this polynomial.
- So, the classifications are: "cubic" and "monomial".
Now let's fill in the table with these classifications:
\begin{tabular}{|c|c|c|}
\hline Polynomial & Degree & Number of Terms \\
\hline [tex]$x^5 + 5x^3 - 2x^2 + 3x$[/tex] & quintic & four terms \\
\hline [tex]$5 - t - 2t^4$[/tex] & quartic & trinomial \\
\hline [tex]$8y - \frac{6y^2}{7^3}$[/tex] & quadratic & binomial \\
\hline [tex]$2x^5 y^3 + 3$[/tex] & 8th degree & binomial \\
\hline [tex]$4m - m^2 + 1$[/tex] & quadratic & trinomial \\
\hline [tex]$-2g^2h$[/tex] & cubic & monomial \\
\hline
\end{tabular}
This table matches each polynomial with the appropriate degree and the number of terms classification.
