Answer :
- Combine like terms by adding the coefficients of terms with the same variable and exponent.
- Add the $x^3$ terms: $7x^3 + 2x^3 = 9x^3$.
- Add the $x^2$ terms: $-4x^2 + (-4x^2) = -8x^2$.
- The sum of the polynomials is $\boxed{9 x^3-8 x^2}$.
### Explanation
1. Understanding the problem
We are asked to find the sum of two polynomials: $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$. This involves combining like terms, which are terms with the same variable and exponent.
2. Combining like terms
To add the polynomials, we combine the terms with $x^3$ and the terms with $x^2$:
$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$
3. Performing the addition
Now, we perform the addition:
$7x^3 + 2x^3 = 9x^3$
$-4x^2 - 4x^2 = -8x^2$
4. Final result
So, the sum of the polynomials is:
$9x^3 - 8x^2$
### Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design bridges and predict their stability. Economists use them to model economic growth, and computer graphics rely on polynomials to create smooth curves and surfaces. Understanding how to add and manipulate polynomials is fundamental to these applications.
- Add the $x^3$ terms: $7x^3 + 2x^3 = 9x^3$.
- Add the $x^2$ terms: $-4x^2 + (-4x^2) = -8x^2$.
- The sum of the polynomials is $\boxed{9 x^3-8 x^2}$.
### Explanation
1. Understanding the problem
We are asked to find the sum of two polynomials: $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$. This involves combining like terms, which are terms with the same variable and exponent.
2. Combining like terms
To add the polynomials, we combine the terms with $x^3$ and the terms with $x^2$:
$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$
3. Performing the addition
Now, we perform the addition:
$7x^3 + 2x^3 = 9x^3$
$-4x^2 - 4x^2 = -8x^2$
4. Final result
So, the sum of the polynomials is:
$9x^3 - 8x^2$
### Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design bridges and predict their stability. Economists use them to model economic growth, and computer graphics rely on polynomials to create smooth curves and surfaces. Understanding how to add and manipulate polynomials is fundamental to these applications.
