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Answer :
- Distribute $(3x+5)$ over $(2x^2 - 3x + 2)$.
- Expand each term: $6x^3 - 9x^2 + 6x + 10x^2 - 15x + 10$.
- Combine like terms: $6x^3 + (-9x^2 + 10x^2) + (6x - 15x) + 10$.
- Simplify to get the final expression: $\boxed{6x^3 + x^2 - 9x + 10}$.
### Explanation
1. Understanding the Problem
We are given the expression $(3 x+5)(2 x^2-3 x+2)$ to multiply and simplify. Our goal is to expand this product and combine like terms to arrive at a simplified polynomial.
2. Distributing the Terms
To multiply the two polynomials, we distribute each term in the first polynomial $(3x + 5)$ to each term in the second polynomial $(2x^2 - 3x + 2)$. This gives us:
$3x(2x^2 - 3x + 2) + 5(2x^2 - 3x + 2)$
3. Expanding the Terms
Now, we expand each term:
$3x(2x^2) + 3x(-3x) + 3x(2) + 5(2x^2) + 5(-3x) + 5(2)$
$= 6x^3 - 9x^2 + 6x + 10x^2 - 15x + 10$
4. Combining Like Terms
Next, we combine like terms. We have the $x^3$ term, the $x^2$ terms, the $x$ terms, and the constant term:
$6x^3 + (-9x^2 + 10x^2) + (6x - 15x) + 10$
$= 6x^3 + x^2 - 9x + 10$
5. Final Result
Therefore, the simplified expression is $6x^3 + x^2 - 9x + 10$.
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, engineers use polynomials to model the behavior of a system, and multiplying these polynomials helps in analyzing the combined effect of different components. In computer graphics, polynomial multiplication is used to perform transformations and rendering of images.
- Expand each term: $6x^3 - 9x^2 + 6x + 10x^2 - 15x + 10$.
- Combine like terms: $6x^3 + (-9x^2 + 10x^2) + (6x - 15x) + 10$.
- Simplify to get the final expression: $\boxed{6x^3 + x^2 - 9x + 10}$.
### Explanation
1. Understanding the Problem
We are given the expression $(3 x+5)(2 x^2-3 x+2)$ to multiply and simplify. Our goal is to expand this product and combine like terms to arrive at a simplified polynomial.
2. Distributing the Terms
To multiply the two polynomials, we distribute each term in the first polynomial $(3x + 5)$ to each term in the second polynomial $(2x^2 - 3x + 2)$. This gives us:
$3x(2x^2 - 3x + 2) + 5(2x^2 - 3x + 2)$
3. Expanding the Terms
Now, we expand each term:
$3x(2x^2) + 3x(-3x) + 3x(2) + 5(2x^2) + 5(-3x) + 5(2)$
$= 6x^3 - 9x^2 + 6x + 10x^2 - 15x + 10$
4. Combining Like Terms
Next, we combine like terms. We have the $x^3$ term, the $x^2$ terms, the $x$ terms, and the constant term:
$6x^3 + (-9x^2 + 10x^2) + (6x - 15x) + 10$
$= 6x^3 + x^2 - 9x + 10$
5. Final Result
Therefore, the simplified expression is $6x^3 + x^2 - 9x + 10$.
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, engineers use polynomials to model the behavior of a system, and multiplying these polynomials helps in analyzing the combined effect of different components. In computer graphics, polynomial multiplication is used to perform transformations and rendering of images.
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