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Sagot :
Let's simplify and classify each polynomial in terms of its standard form and degree.
### Process
1. Simplify each polynomial to its standard form:
- Combine like terms.
- Arrange the terms in descending order according to the degree of [tex]\( x \)[/tex].
2. Classify each polynomial based on its degree:
- Degree 0: Constant polynomial
- Degree 1: Linear polynomial
- Degree 2: Quadratic polynomial
- Degree 3: Cubic polynomial
- Degree 4: Quartic polynomial
- Degree 5: Quintic polynomial
### Simplification and Classification
1. [tex]\( 9 + 2x^3 - 7x^3 - 2x^3 \)[/tex]
- Simplify: [tex]\( 9 + (2 - 7 - 2)x^3 = 9 - 7x^3 \)[/tex]
- Standard form: [tex]\( 9 - 7x^3 \)[/tex]
- Degree: 3 (Cubic)
2. [tex]\( -8 + 9x^5 + 8 \)[/tex]
- Simplify: [tex]\( (-8 + 8) + 9x^5 = 9x^5 \)[/tex]
- Standard form: [tex]\( 9x^5 \)[/tex]
- Degree: 5 (Quintic)
3. [tex]\( 7 \)[/tex]
- Simplify: [tex]\( 7 \)[/tex]
- Standard form: [tex]\( 7 \)[/tex]
- Degree: 0 (Constant)
4. [tex]\( -8 - 5x^3 + 4x^5 - 4x - 6 \)[/tex]
- Simplify: [tex]\( (-8 - 6) + 4x^5 - 5x^3 - 4x = -14 + 4x^5 - 5x^3 - 4x \)[/tex]
- Standard form: [tex]\( 4x^5 - 5x^3 - 4x - 14 \)[/tex]
- Degree: 5 (Quintic)
5. [tex]\( -3 - 8x^3 - 6x \)[/tex]
- Simplify: [tex]\( -3 - 8x^3 - 6x \)[/tex]
- Standard form: [tex]\( -8x^3 - 6x - 3 \)[/tex]
- Degree: 3 (Cubic)
6. [tex]\( 4 + 8x^4 \)[/tex]
- Simplify: [tex]\( 4 + 8x^4 \)[/tex]
- Standard form: [tex]\( 8x^4 + 4 \)[/tex]
- Degree: 4 (Quartic)
7. [tex]\( -8 + 7x^4 \)[/tex]
- Simplify: [tex]\( -8 + 7x^4 \)[/tex]
- Standard form: [tex]\( 7x^4 - 8 \)[/tex]
- Degree: 4 (Quartic)
8. [tex]\( -3x^3 \)[/tex]
- Simplify: [tex]\( -3x^3 \)[/tex]
- Standard form: [tex]\( -3x^3 \)[/tex]
- Degree: 3 (Cubic)
9. [tex]\( 8 \)[/tex]
- Simplify: [tex]\( 8 \)[/tex]
- Standard form: [tex]\( 8 \)[/tex]
- Degree: 0 (Constant)
10. [tex]\( 5 - 8x \)[/tex]
- Simplify: [tex]\( 5 - 8x \)[/tex]
- Standard form: [tex]\( -8x + 5 \)[/tex]
- Degree: 1 (Linear)
11. [tex]\( 5 + 7x^5 \)[/tex]
- Simplify: [tex]\( 5 + 7x^5 \)[/tex]
- Standard form: [tex]\( 7x^5 + 5 \)[/tex]
- Degree: 5 (Quintic)
12. [tex]\( 5 \)[/tex]
- Simplify: [tex]\( 5 \)[/tex]
- Standard form: [tex]\( 5 \)[/tex]
- Degree: 0 (Constant)
13. [tex]\( -9x^5 - 7x \)[/tex]
- Simplify: [tex]\( -9x^5 - 7x \)[/tex]
- Standard form: [tex]\( -9x^5 - 7x \)[/tex]
- Degree: 5 (Quintic)
14. [tex]\( 4x + x \)[/tex]
- Simplify: [tex]\( 4x + x = 5x \)[/tex]
- Standard form: [tex]\( 5x \)[/tex]
- Degree: 1 (Linear)
15. [tex]\( 9 - 3x^5 - 2x^2 \)[/tex]
- Simplify: [tex]\( 9 - 3x^5 - 2x^2 \)[/tex]
- Standard form: [tex]\( -3x^5 - 2x^2 + 9 \)[/tex]
- Degree: 5 (Quintic)
16. [tex]\( -x^2 \)[/tex]
- Simplify: [tex]\( -x^2 \)[/tex]
- Standard form: [tex]\( -x^2 \)[/tex]
- Degree: 2 (Quadratic)
17. [tex]\( 4x^3 + 9x^4 + 7 \)[/tex]
- Simplify: [tex]\( 4x^3 + 9x^4 + 7 \)[/tex]
- Standard form: [tex]\( 9x^4 + 4x^3 + 7 \)[/tex]
- Degree: 4 (Quartic)
18. [tex]\( 2 \)[/tex]
- Simplify: [tex]\( 2 \)[/tex]
- Standard form: [tex]\( 2 \)[/tex]
- Degree: 0 (Constant)
19. [tex]\( -9x^4 + 7x^4 - 8 \)[/tex]
- Simplify: [tex]\( (-9x^4 + 7x^4) - 8 = -2x^4 - 8 \)[/tex]
- Standard form: [tex]\( -2x^4 - 8 \)[/tex]
- Degree: 4 (Quartic)
20. [tex]\( 3 - 4x^3 + 1 \)[/tex]
- Simplify: [tex]\( 3 + 1 - 4x^3 = 4 - 4x^3 \)[/tex]
- Standard form: [tex]\( -4x^3 + 4 \)[/tex]
- Degree: 3 (Cubic)
This way, each polynomial is simplified and classified according to its degree.
### Process
1. Simplify each polynomial to its standard form:
- Combine like terms.
- Arrange the terms in descending order according to the degree of [tex]\( x \)[/tex].
2. Classify each polynomial based on its degree:
- Degree 0: Constant polynomial
- Degree 1: Linear polynomial
- Degree 2: Quadratic polynomial
- Degree 3: Cubic polynomial
- Degree 4: Quartic polynomial
- Degree 5: Quintic polynomial
### Simplification and Classification
1. [tex]\( 9 + 2x^3 - 7x^3 - 2x^3 \)[/tex]
- Simplify: [tex]\( 9 + (2 - 7 - 2)x^3 = 9 - 7x^3 \)[/tex]
- Standard form: [tex]\( 9 - 7x^3 \)[/tex]
- Degree: 3 (Cubic)
2. [tex]\( -8 + 9x^5 + 8 \)[/tex]
- Simplify: [tex]\( (-8 + 8) + 9x^5 = 9x^5 \)[/tex]
- Standard form: [tex]\( 9x^5 \)[/tex]
- Degree: 5 (Quintic)
3. [tex]\( 7 \)[/tex]
- Simplify: [tex]\( 7 \)[/tex]
- Standard form: [tex]\( 7 \)[/tex]
- Degree: 0 (Constant)
4. [tex]\( -8 - 5x^3 + 4x^5 - 4x - 6 \)[/tex]
- Simplify: [tex]\( (-8 - 6) + 4x^5 - 5x^3 - 4x = -14 + 4x^5 - 5x^3 - 4x \)[/tex]
- Standard form: [tex]\( 4x^5 - 5x^3 - 4x - 14 \)[/tex]
- Degree: 5 (Quintic)
5. [tex]\( -3 - 8x^3 - 6x \)[/tex]
- Simplify: [tex]\( -3 - 8x^3 - 6x \)[/tex]
- Standard form: [tex]\( -8x^3 - 6x - 3 \)[/tex]
- Degree: 3 (Cubic)
6. [tex]\( 4 + 8x^4 \)[/tex]
- Simplify: [tex]\( 4 + 8x^4 \)[/tex]
- Standard form: [tex]\( 8x^4 + 4 \)[/tex]
- Degree: 4 (Quartic)
7. [tex]\( -8 + 7x^4 \)[/tex]
- Simplify: [tex]\( -8 + 7x^4 \)[/tex]
- Standard form: [tex]\( 7x^4 - 8 \)[/tex]
- Degree: 4 (Quartic)
8. [tex]\( -3x^3 \)[/tex]
- Simplify: [tex]\( -3x^3 \)[/tex]
- Standard form: [tex]\( -3x^3 \)[/tex]
- Degree: 3 (Cubic)
9. [tex]\( 8 \)[/tex]
- Simplify: [tex]\( 8 \)[/tex]
- Standard form: [tex]\( 8 \)[/tex]
- Degree: 0 (Constant)
10. [tex]\( 5 - 8x \)[/tex]
- Simplify: [tex]\( 5 - 8x \)[/tex]
- Standard form: [tex]\( -8x + 5 \)[/tex]
- Degree: 1 (Linear)
11. [tex]\( 5 + 7x^5 \)[/tex]
- Simplify: [tex]\( 5 + 7x^5 \)[/tex]
- Standard form: [tex]\( 7x^5 + 5 \)[/tex]
- Degree: 5 (Quintic)
12. [tex]\( 5 \)[/tex]
- Simplify: [tex]\( 5 \)[/tex]
- Standard form: [tex]\( 5 \)[/tex]
- Degree: 0 (Constant)
13. [tex]\( -9x^5 - 7x \)[/tex]
- Simplify: [tex]\( -9x^5 - 7x \)[/tex]
- Standard form: [tex]\( -9x^5 - 7x \)[/tex]
- Degree: 5 (Quintic)
14. [tex]\( 4x + x \)[/tex]
- Simplify: [tex]\( 4x + x = 5x \)[/tex]
- Standard form: [tex]\( 5x \)[/tex]
- Degree: 1 (Linear)
15. [tex]\( 9 - 3x^5 - 2x^2 \)[/tex]
- Simplify: [tex]\( 9 - 3x^5 - 2x^2 \)[/tex]
- Standard form: [tex]\( -3x^5 - 2x^2 + 9 \)[/tex]
- Degree: 5 (Quintic)
16. [tex]\( -x^2 \)[/tex]
- Simplify: [tex]\( -x^2 \)[/tex]
- Standard form: [tex]\( -x^2 \)[/tex]
- Degree: 2 (Quadratic)
17. [tex]\( 4x^3 + 9x^4 + 7 \)[/tex]
- Simplify: [tex]\( 4x^3 + 9x^4 + 7 \)[/tex]
- Standard form: [tex]\( 9x^4 + 4x^3 + 7 \)[/tex]
- Degree: 4 (Quartic)
18. [tex]\( 2 \)[/tex]
- Simplify: [tex]\( 2 \)[/tex]
- Standard form: [tex]\( 2 \)[/tex]
- Degree: 0 (Constant)
19. [tex]\( -9x^4 + 7x^4 - 8 \)[/tex]
- Simplify: [tex]\( (-9x^4 + 7x^4) - 8 = -2x^4 - 8 \)[/tex]
- Standard form: [tex]\( -2x^4 - 8 \)[/tex]
- Degree: 4 (Quartic)
20. [tex]\( 3 - 4x^3 + 1 \)[/tex]
- Simplify: [tex]\( 3 + 1 - 4x^3 = 4 - 4x^3 \)[/tex]
- Standard form: [tex]\( -4x^3 + 4 \)[/tex]
- Degree: 3 (Cubic)
This way, each polynomial is simplified and classified according to its degree.
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