Certainly! Let's classify each polynomial based on its degree and number of terms according to the steps provided:
1. Polynomial: \(x^5 + 5x^3 - 2x^2 + 3x\)
- Degree: 5 (The highest power of the variable \(x\))
- Number of Terms: 4 (Separated by plus and minus signs)
- Classification:
- Quintic (degree 5)
- Four terms (4 terms)
2. Polynomial: \(5 - t - 2t^4\)
- Degree: 4 (The highest power of the variable \(t\))
- Number of Terms: 3 (Separated by plus and minus signs)
- Classification:
- Quartic (degree 4)
- Trinomial (3 terms)
3. Polynomial: \(8y - \frac{6y^2}{7^3}\)
- Degree: 2 (The highest power of the variable \(y\))
- Number of Terms: 2 (Separated by plus and minus signs)
- Classification:
- Quadratic (degree 2)
- Binomial (2 terms)
4. Polynomial: \(2x^5y^3 + 3\)
- Degree: 8 (The sum of the highest powers of \(x\) and \(y\) in the term \(2x^5y^3\))
- Number of Terms: 2 (Separated by plus and minus signs)
- Classification:
- Eighth-degree polynomial (degree 8)
- Binomial (2 terms)
5. Polynomial: \(4m - m^2 + 1\)
- Degree: 2 (The highest power of the variable \(m\))
- Number of Terms: 3 (Separated by plus and minus signs)
- Classification:
- Quadratic (degree 2)
- Trinomial (3 terms)
6. Polynomial: \(-2g^2h\)
- Degree: 3 (The sum of the highest powers of \(g\) and \(h\) in the term \(-2g^2h\))
- Number of Terms: 1 (Since it is a single term)
- Classification:
- Cubic (degree 3)
- Monomial (1 term)
Summary Table:
\begin{tabular}{|c|c|c|}
\hline Polynomial & Degree & Number of Terms \\
\hline [tex]$x^5+5 x^3-2 x^2+3 x$[/tex] & Quintic (5) & Four terms (4) \\
\hline [tex]$5-t-2 t^4$[/tex] & Quartic (4) & Trinomial (3) \\
\hline [tex]$8 y-\frac{6 y^2}{7^3}$[/tex] & Quadratic (2) & Binomial (2) \\
\hline [tex]$2 x^5 y^3+3$[/tex] & Eighth-degree (8) & Binomial (2) \\
\hline [tex]$4 m-m^2+1$[/tex] & Quadratic (2) & Trinomial (3) \\
\hline[tex]$-2 g^2 h$[/tex] & Cubic (3) & Monomial (1) \\
\hline
\end{tabular}
This classification provides a clear understanding of each polynomial's degree and number of terms.
