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Answer :
- Add the polynomials by combining like terms: $(7x^3 - 4x^2) + (2x^3 - 4x^2)$.
- Combine the $x^3$ terms: $(7+2)x^3 = 9x^3$.
- Combine the $x^2$ terms: $(-4-4)x^2 = -8x^2$.
- The sum of the polynomials is $\boxed{9x^3 - 8x^2}$.
### Explanation
1. Problem Analysis
We are asked to find the sum of the polynomials $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$. To do this, we combine like terms.
2. Combining Like Terms
We add the two polynomials by combining like terms:
$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$
3. Simplifying the Expression
Now, we simplify the expression by performing the addition of the coefficients:
$(7+2)x^3 + (-4-4)x^2 = 9x^3 + (-8)x^2 = 9x^3 - 8x^2$
4. Final Answer
Therefore, the sum of the polynomials is $9x^3 - 8x^2$.
### Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, computer graphics, and economics. For example, engineers use polynomials to design bridges and buildings, while economists use them to model economic growth. Understanding how to add polynomials is fundamental to these applications.
- Combine the $x^3$ terms: $(7+2)x^3 = 9x^3$.
- Combine the $x^2$ terms: $(-4-4)x^2 = -8x^2$.
- The sum of the polynomials is $\boxed{9x^3 - 8x^2}$.
### Explanation
1. Problem Analysis
We are asked to find the sum of the polynomials $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$. To do this, we combine like terms.
2. Combining Like Terms
We add the two polynomials by combining like terms:
$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$
3. Simplifying the Expression
Now, we simplify the expression by performing the addition of the coefficients:
$(7+2)x^3 + (-4-4)x^2 = 9x^3 + (-8)x^2 = 9x^3 - 8x^2$
4. Final Answer
Therefore, the sum of the polynomials is $9x^3 - 8x^2$.
### Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, computer graphics, and economics. For example, engineers use polynomials to design bridges and buildings, while economists use them to model economic growth. Understanding how to add polynomials is fundamental to these applications.
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