Answer :
- Combine like terms: Add 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 given two polynomials, $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$, and we want to find their sum. This involves combining like terms, which are terms with the same variable raised to the same power.
2. Combining Like Terms
To add the polynomials, we combine the $x^3$ terms and the $x^2$ terms separately:
$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$
3. Performing the Addition
Now, we perform the addition:
$7x^3 + 2x^3 = (7+2)x^3 = 9x^3$
$-4x^2 - 4x^2 = (-4-4)x^2 = -8x^2$
So, the sum is $9x^3 - 8x^2$.
4. Final Answer
Therefore, the sum of the polynomials $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$ is $9x^3 - 8x^2$.
### Examples
Polynomials are used in various fields like physics, engineering, and economics to model complex relationships. For example, in physics, polynomials can describe the trajectory of a projectile, while in economics, they can model cost and revenue functions. Understanding how to add and manipulate polynomials is fundamental to solving problems in these areas.
- 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 given two polynomials, $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$, and we want to find their sum. This involves combining like terms, which are terms with the same variable raised to the same power.
2. Combining Like Terms
To add the polynomials, we combine the $x^3$ terms and the $x^2$ terms separately:
$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$
3. Performing the Addition
Now, we perform the addition:
$7x^3 + 2x^3 = (7+2)x^3 = 9x^3$
$-4x^2 - 4x^2 = (-4-4)x^2 = -8x^2$
So, the sum is $9x^3 - 8x^2$.
4. Final Answer
Therefore, the sum of the polynomials $(7x^3 - 4x^2)$ and $(2x^3 - 4x^2)$ is $9x^3 - 8x^2$.
### Examples
Polynomials are used in various fields like physics, engineering, and economics to model complex relationships. For example, in physics, polynomials can describe the trajectory of a projectile, while in economics, they can model cost and revenue functions. Understanding how to add and manipulate polynomials is fundamental to solving problems in these areas.
