Answer :
Sure, let's carefully analyze each of the given expressions step-by-step, ensuring we understand each term within the context of algebraic manipulation.
### Expression 3: [tex]\( 5t^3 - 15x + 20t \)[/tex]
1. Term Analysis:
- [tex]\(5t^3\)[/tex]: This term involves [tex]\(t\)[/tex] raised to the power of 3 and is multiplied by 5.
- [tex]\(-15x\)[/tex]: This term involves [tex]\(x\)[/tex] and is multiplied by -15.
- [tex]\(20t\)[/tex]: This term involves [tex]\(t\)[/tex] and is multiplied by 20.
2. Aggregation:
- The expression doesn't have terms that need combining since they involve different variables and powers.
### Expression 4: [tex]\( x^5 - 4x^2 - 34x \)[/tex]
1. Term Analysis:
- [tex]\(x^5\)[/tex]: This term is [tex]\(x\)[/tex] raised to the power of 5.
- [tex]\(-4x^2\)[/tex]: This term is [tex]\(x\)[/tex] squared and multiplied by -4.
- [tex]\(-34x\)[/tex]: This term is [tex]\(x\)[/tex] and multiplied by -34.
2. Aggregation:
- Similar to the previous expressions, the terms involve different powers of [tex]\(x\)[/tex] and cannot be combined further.
### Expression 2: [tex]\( 35x^5y^2 + 21x^4y^2 + 14x^3y^3 \)[/tex]
1. Term Analysis:
- [tex]\(35x^5y^2\)[/tex]: This term involves [tex]\(x\)[/tex] raised to the power of 5 and [tex]\(y\)[/tex] squared, multiplied by 35.
- [tex]\(21x^4y^2\)[/tex]: This term involves [tex]\(x\)[/tex] raised to the power of 4 and [tex]\(y\)[/tex] squared, multiplied by 21.
- [tex]\(14x^3y^3\)[/tex]: This term involves [tex]\(x\)[/tex] raised to the power of 3 and [tex]\(y\)[/tex] raised to the power of 3, multiplied by 14.
2. Aggregation:
- The terms involve different powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], thus they cannot be simplified further individually.
### Combined Result:
Now, assuming the order specified in the list, the ordering of our results should be:
1. [tex]\( 5t^3 - 15x + 20t \)[/tex]
2. [tex]\( 35x^5y^2 + 21x^4y^2 + 14x^3y^3 \)[/tex]
3. [tex]\( x^5 - 4x^2 - 34x \)[/tex]
Here is a recap of each expression in the given order:
1. [tex]\( 5t^3 - 15x + 20t \)[/tex]
2. [tex]\( 35x^5y^2 + 21x^4y^2 + 14x^3y^3 \)[/tex]
3. [tex]\( x^5 - 4x^2 - 34x \)[/tex]
These represent the final expressions as detailed above. Each term retains its structure because there are no like terms to combine within each expression.
### Expression 3: [tex]\( 5t^3 - 15x + 20t \)[/tex]
1. Term Analysis:
- [tex]\(5t^3\)[/tex]: This term involves [tex]\(t\)[/tex] raised to the power of 3 and is multiplied by 5.
- [tex]\(-15x\)[/tex]: This term involves [tex]\(x\)[/tex] and is multiplied by -15.
- [tex]\(20t\)[/tex]: This term involves [tex]\(t\)[/tex] and is multiplied by 20.
2. Aggregation:
- The expression doesn't have terms that need combining since they involve different variables and powers.
### Expression 4: [tex]\( x^5 - 4x^2 - 34x \)[/tex]
1. Term Analysis:
- [tex]\(x^5\)[/tex]: This term is [tex]\(x\)[/tex] raised to the power of 5.
- [tex]\(-4x^2\)[/tex]: This term is [tex]\(x\)[/tex] squared and multiplied by -4.
- [tex]\(-34x\)[/tex]: This term is [tex]\(x\)[/tex] and multiplied by -34.
2. Aggregation:
- Similar to the previous expressions, the terms involve different powers of [tex]\(x\)[/tex] and cannot be combined further.
### Expression 2: [tex]\( 35x^5y^2 + 21x^4y^2 + 14x^3y^3 \)[/tex]
1. Term Analysis:
- [tex]\(35x^5y^2\)[/tex]: This term involves [tex]\(x\)[/tex] raised to the power of 5 and [tex]\(y\)[/tex] squared, multiplied by 35.
- [tex]\(21x^4y^2\)[/tex]: This term involves [tex]\(x\)[/tex] raised to the power of 4 and [tex]\(y\)[/tex] squared, multiplied by 21.
- [tex]\(14x^3y^3\)[/tex]: This term involves [tex]\(x\)[/tex] raised to the power of 3 and [tex]\(y\)[/tex] raised to the power of 3, multiplied by 14.
2. Aggregation:
- The terms involve different powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], thus they cannot be simplified further individually.
### Combined Result:
Now, assuming the order specified in the list, the ordering of our results should be:
1. [tex]\( 5t^3 - 15x + 20t \)[/tex]
2. [tex]\( 35x^5y^2 + 21x^4y^2 + 14x^3y^3 \)[/tex]
3. [tex]\( x^5 - 4x^2 - 34x \)[/tex]
Here is a recap of each expression in the given order:
1. [tex]\( 5t^3 - 15x + 20t \)[/tex]
2. [tex]\( 35x^5y^2 + 21x^4y^2 + 14x^3y^3 \)[/tex]
3. [tex]\( x^5 - 4x^2 - 34x \)[/tex]
These represent the final expressions as detailed above. Each term retains its structure because there are no like terms to combine within each expression.
