Answer :
- Add the polynomials by combining like terms: $(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$.
- Simplify the $x^3$ terms: $7x^3 + 2x^3 = 9x^3$.
- Simplify the $x^2$ terms: $-4x^2 - 4x^2 = -8x^2$.
- The sum of the polynomials is $\boxed{9x^3 - 8x^2}$.
### 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.
2. Combining like terms
To add the polynomials, we combine the terms with the same exponent of $x$. So, we add the $x^3$ terms together and the $x^2$ terms together:
$$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$$
3. Performing the addition
Now, we perform the addition:
$$7x^3 + 2x^3 = 9x^3$$
$$-4x^2 - 4x^2 = -8x^2$$
So, the sum of the polynomials is $9x^3 - 8x^2$.
4. Final Answer
Therefore, the sum of the given polynomials is $9x^3 - 8x^2$.
### Examples
Polynomials are used in many areas of mathematics and science. For example, they can be used to model the trajectory of a ball thrown in the air, or to describe the shape of a curve. In economics, polynomials can be used to represent cost and revenue functions, helping businesses analyze their profitability. Understanding how to add polynomials is a fundamental skill that enables us to solve a wide range of real-world problems.
- Simplify the $x^3$ terms: $7x^3 + 2x^3 = 9x^3$.
- Simplify the $x^2$ terms: $-4x^2 - 4x^2 = -8x^2$.
- The sum of the polynomials is $\boxed{9x^3 - 8x^2}$.
### 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.
2. Combining like terms
To add the polynomials, we combine the terms with the same exponent of $x$. So, we add the $x^3$ terms together and the $x^2$ terms together:
$$(7x^3 - 4x^2) + (2x^3 - 4x^2) = (7x^3 + 2x^3) + (-4x^2 - 4x^2)$$
3. Performing the addition
Now, we perform the addition:
$$7x^3 + 2x^3 = 9x^3$$
$$-4x^2 - 4x^2 = -8x^2$$
So, the sum of the polynomials is $9x^3 - 8x^2$.
4. Final Answer
Therefore, the sum of the given polynomials is $9x^3 - 8x^2$.
### Examples
Polynomials are used in many areas of mathematics and science. For example, they can be used to model the trajectory of a ball thrown in the air, or to describe the shape of a curve. In economics, polynomials can be used to represent cost and revenue functions, helping businesses analyze their profitability. Understanding how to add polynomials is a fundamental skill that enables us to solve a wide range of real-world problems.
