Answer :
Let's go through the expression [tex]\(14x^5 + 35x^3 - 4x^2 - 10\)[/tex].
### Understanding the Expression:
1. Identify Terms and Their Degrees:
- [tex]\(14x^5\)[/tex]: This is the highest degree term, where the exponent of [tex]\(x\)[/tex] is 5.
- [tex]\(35x^3\)[/tex]: This is the next term, with [tex]\(x\)[/tex] raised to the 3rd power.
- [tex]\(-4x^2\)[/tex]: This term involves [tex]\(x\)[/tex] to the 2nd power.
- [tex]\(-10\)[/tex]: This is a constant term, meaning it doesn't involve the variable [tex]\(x\)[/tex].
2. Structure of the Polynomial:
- The polynomial is written in descending order of the powers of [tex]\(x\)[/tex]. This standard form makes it easier to analyze or perform operations like differentiation, integration, or factoring.
3. Analyzing the Coefficients:
- Each term has a coefficient which is the numerical part in front of the variable:
- Coefficient of [tex]\(x^5\)[/tex] is 14.
- Coefficient of [tex]\(x^3\)[/tex] is 35.
- Coefficient of [tex]\(x^2\)[/tex] is -4.
- The constant term is -10.
### Key Concepts for Understanding or Solving the Polynomial:
- Operations you can perform:
- Addition/Subtraction: Combine like terms if there are any similar powers of [tex]\(x\)[/tex].
- Factorization: Attempt to rewrite the polynomial as a product of its factors. This is most useful if solving for when the polynomial equals zero.
- Differentiation & Integration:
- If tasked with differentiating or integrating, apply the respective rules to each term separately based on their powers.
### Conclusion:
This polynomial is a quintic function (degree 5) and does not seem to have like terms to combine. It is left as is unless additional operations or a specific purpose (like solving for [tex]\(x\)[/tex] when the expression equals zero) are required. This setup is helpful for application in calculus or algebraic operations!
### Understanding the Expression:
1. Identify Terms and Their Degrees:
- [tex]\(14x^5\)[/tex]: This is the highest degree term, where the exponent of [tex]\(x\)[/tex] is 5.
- [tex]\(35x^3\)[/tex]: This is the next term, with [tex]\(x\)[/tex] raised to the 3rd power.
- [tex]\(-4x^2\)[/tex]: This term involves [tex]\(x\)[/tex] to the 2nd power.
- [tex]\(-10\)[/tex]: This is a constant term, meaning it doesn't involve the variable [tex]\(x\)[/tex].
2. Structure of the Polynomial:
- The polynomial is written in descending order of the powers of [tex]\(x\)[/tex]. This standard form makes it easier to analyze or perform operations like differentiation, integration, or factoring.
3. Analyzing the Coefficients:
- Each term has a coefficient which is the numerical part in front of the variable:
- Coefficient of [tex]\(x^5\)[/tex] is 14.
- Coefficient of [tex]\(x^3\)[/tex] is 35.
- Coefficient of [tex]\(x^2\)[/tex] is -4.
- The constant term is -10.
### Key Concepts for Understanding or Solving the Polynomial:
- Operations you can perform:
- Addition/Subtraction: Combine like terms if there are any similar powers of [tex]\(x\)[/tex].
- Factorization: Attempt to rewrite the polynomial as a product of its factors. This is most useful if solving for when the polynomial equals zero.
- Differentiation & Integration:
- If tasked with differentiating or integrating, apply the respective rules to each term separately based on their powers.
### Conclusion:
This polynomial is a quintic function (degree 5) and does not seem to have like terms to combine. It is left as is unless additional operations or a specific purpose (like solving for [tex]\(x\)[/tex] when the expression equals zero) are required. This setup is helpful for application in calculus or algebraic operations!
