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Answer :
- Expand the product of the binomial and trinomial: $(4x - 3)(3x^2 - 4x - 3)$.
- Distribute and multiply each term: $12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9$.
- Combine like terms: $12x^3 - 25x^2 + 9$.
- The correct simplification is $\boxed{12 x^3-25 x^2+9}$.
### Explanation
1. Understanding the Problem
We are given the expression $(4x - 3)(3x^2 - 4x - 3)$ and four possible simplifications. Our objective is to choose the correct simplification from the given options. The expression involves multiplying a binomial by a trinomial. We need to expand the product and combine like terms to simplify the expression.
2. Expanding the Product
To simplify the expression $(4x - 3)(3x^2 - 4x - 3)$, we need to multiply each term in the first factor by each term in the second factor. This is also known as expanding the product.
3. Applying the Distributive Property
Expanding the product, we get:
$4x(3x^2 - 4x - 3) - 3(3x^2 - 4x - 3)$
4. Performing the Multiplication
Now, we distribute $4x$ and $-3$ to each term in the parentheses:
$4x(3x^2) + 4x(-4x) + 4x(-3) - 3(3x^2) - 3(-4x) - 3(-3)$
$= 12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9$
5. Combining Like Terms
Next, we combine like terms:
$12x^3 + (-16x^2 - 9x^2) + (-12x + 12x) + 9$
$= 12x^3 - 25x^2 + 0x + 9$
$= 12x^3 - 25x^2 + 9$
6. Identifying the Correct Option
Comparing the simplified expression $12x^3 - 25x^2 + 9$ with the given options, we find that it matches option d.
Therefore, the correct simplification is $12x^3 - 25x^2 + 9$.
### Examples
Understanding polynomial multiplication is essential in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when analyzing the trajectory of a projectile, engineers use polynomial functions to describe its path, and simplifying these functions helps in predicting the projectile's landing point or optimizing its launch angle. Similarly, in economics, polynomial functions can model cost and revenue curves, and simplifying these functions aids in determining break-even points or maximizing profits.
- Distribute and multiply each term: $12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9$.
- Combine like terms: $12x^3 - 25x^2 + 9$.
- The correct simplification is $\boxed{12 x^3-25 x^2+9}$.
### Explanation
1. Understanding the Problem
We are given the expression $(4x - 3)(3x^2 - 4x - 3)$ and four possible simplifications. Our objective is to choose the correct simplification from the given options. The expression involves multiplying a binomial by a trinomial. We need to expand the product and combine like terms to simplify the expression.
2. Expanding the Product
To simplify the expression $(4x - 3)(3x^2 - 4x - 3)$, we need to multiply each term in the first factor by each term in the second factor. This is also known as expanding the product.
3. Applying the Distributive Property
Expanding the product, we get:
$4x(3x^2 - 4x - 3) - 3(3x^2 - 4x - 3)$
4. Performing the Multiplication
Now, we distribute $4x$ and $-3$ to each term in the parentheses:
$4x(3x^2) + 4x(-4x) + 4x(-3) - 3(3x^2) - 3(-4x) - 3(-3)$
$= 12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9$
5. Combining Like Terms
Next, we combine like terms:
$12x^3 + (-16x^2 - 9x^2) + (-12x + 12x) + 9$
$= 12x^3 - 25x^2 + 0x + 9$
$= 12x^3 - 25x^2 + 9$
6. Identifying the Correct Option
Comparing the simplified expression $12x^3 - 25x^2 + 9$ with the given options, we find that it matches option d.
Therefore, the correct simplification is $12x^3 - 25x^2 + 9$.
### Examples
Understanding polynomial multiplication is essential in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when analyzing the trajectory of a projectile, engineers use polynomial functions to describe its path, and simplifying these functions helps in predicting the projectile's landing point or optimizing its launch angle. Similarly, in economics, polynomial functions can model cost and revenue curves, and simplifying these functions aids in determining break-even points or maximizing profits.
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