Chapter5: Polynomials And Polynomial Functions
Section5.1: Add And Subtract Polynomials
Problem 78E: Using your own words, explain the difference between a polynomial with five terms and a polynomial...
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Factor completely
![### Polynomial Equation Example
In this example, we are given a polynomial equation which is as follows:
\[ 8x^6 + 6x^3 + 20x^3 + 15 \]
Here is a breakdown of each term within the equation:
1. **\(8x^6\)**: This term represents 8 times \(x\) raised to the power of 6.
2. **\(6x^3\)**: This term denotes 6 times \(x\) raised to the power of 3.
3. **\(20x^3\)**: This term signifies 20 times \(x\) raised to the power of 3.
4. **15**: This is a constant term without any variable attached to it.
### Key Points to Note:
- **Degree of the Polynomial**: The highest power of the variable \(x\) in the polynomial is 6 (from the term \(8x^6\)). Therefore, this is a polynomial of degree 6.
- **Like Terms**: Notice that \(6x^3\) and \(20x^3\) are like terms, and they can be combined to simplify the polynomial further.
To simplify the polynomial:
\[ 8x^6 + (6x^3 + 20x^3) + 15 \]
\[ 8x^6 + 26x^3 + 15 \]
This produces a more simplified version of the polynomial equation. This summary is useful for understanding the composition and simplification of polynomial expressions, which is a fundamental concept in algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a79e915-3d5c-4c79-9ba5-7a5ef81967b7%2F19485478-df66-4c32-97c4-ee94b70c5c7a%2Fav1uhn_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Equation Example
In this example, we are given a polynomial equation which is as follows:
\[ 8x^6 + 6x^3 + 20x^3 + 15 \]
Here is a breakdown of each term within the equation:
1. **\(8x^6\)**: This term represents 8 times \(x\) raised to the power of 6.
2. **\(6x^3\)**: This term denotes 6 times \(x\) raised to the power of 3.
3. **\(20x^3\)**: This term signifies 20 times \(x\) raised to the power of 3.
4. **15**: This is a constant term without any variable attached to it.
### Key Points to Note:
- **Degree of the Polynomial**: The highest power of the variable \(x\) in the polynomial is 6 (from the term \(8x^6\)). Therefore, this is a polynomial of degree 6.
- **Like Terms**: Notice that \(6x^3\) and \(20x^3\) are like terms, and they can be combined to simplify the polynomial further.
To simplify the polynomial:
\[ 8x^6 + (6x^3 + 20x^3) + 15 \]
\[ 8x^6 + 26x^3 + 15 \]
This produces a more simplified version of the polynomial equation. This summary is useful for understanding the composition and simplification of polynomial expressions, which is a fundamental concept in algebra.
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