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Answer :
- Distribute $(x^2+3x+9)(x-3)$ as $x^2(x-3) + 3x(x-3) + 9(x-3)$.
- Expand each term: $x^3 - 3x^2 + 3x^2 - 9x + 9x - 27$.
- Combine like terms: $x^3 - 27$.
- The product is $\boxed{x^3-27}$.
### Explanation
1. Understanding the Problem
We are given the expression $(x^2+3x+9)(x-3)$ and asked to find its product. The question provides four possible answers: $x^3-27$, $x^2+4x+6$, $x^3-6x^2-18x-27$, and $-6x^4+x^3-18x^2-27$
2. Setting up the Multiplication
To find the product of $(x^2+3x+9)$ and $(x-3)$, we need to multiply the two polynomials. We can use the distributive property to do this:
$(x^2+3x+9)(x-3) = x^2(x-3) + 3x(x-3) + 9(x-3)$
3. Distributing the Terms
Now, we distribute each term:
$x^2(x-3) = x^3 - 3x^2$
$3x(x-3) = 3x^2 - 9x$
$9(x-3) = 9x - 27$
4. Combining the Terms
Next, we add the results together:
$(x^3 - 3x^2) + (3x^2 - 9x) + (9x - 27) = x^3 - 3x^2 + 3x^2 - 9x + 9x - 27$
5. Simplifying the Expression
Now, we simplify by combining like terms:
$x^3 - 3x^2 + 3x^2 - 9x + 9x - 27 = x^3 + (-3x^2 + 3x^2) + (-9x + 9x) - 27 = x^3 - 27$
6. Finding the Correct Answer
The product of $(x^2+3x+9)$ and $(x-3)$ is $x^3 - 27$. Comparing this with the given options, we see that option 1 is the correct answer.
### Examples
Polynomial multiplication, like in this problem, is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and stress on different parts of the structure. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces for 3D models.
- Expand each term: $x^3 - 3x^2 + 3x^2 - 9x + 9x - 27$.
- Combine like terms: $x^3 - 27$.
- The product is $\boxed{x^3-27}$.
### Explanation
1. Understanding the Problem
We are given the expression $(x^2+3x+9)(x-3)$ and asked to find its product. The question provides four possible answers: $x^3-27$, $x^2+4x+6$, $x^3-6x^2-18x-27$, and $-6x^4+x^3-18x^2-27$
2. Setting up the Multiplication
To find the product of $(x^2+3x+9)$ and $(x-3)$, we need to multiply the two polynomials. We can use the distributive property to do this:
$(x^2+3x+9)(x-3) = x^2(x-3) + 3x(x-3) + 9(x-3)$
3. Distributing the Terms
Now, we distribute each term:
$x^2(x-3) = x^3 - 3x^2$
$3x(x-3) = 3x^2 - 9x$
$9(x-3) = 9x - 27$
4. Combining the Terms
Next, we add the results together:
$(x^3 - 3x^2) + (3x^2 - 9x) + (9x - 27) = x^3 - 3x^2 + 3x^2 - 9x + 9x - 27$
5. Simplifying the Expression
Now, we simplify by combining like terms:
$x^3 - 3x^2 + 3x^2 - 9x + 9x - 27 = x^3 + (-3x^2 + 3x^2) + (-9x + 9x) - 27 = x^3 - 27$
6. Finding the Correct Answer
The product of $(x^2+3x+9)$ and $(x-3)$ is $x^3 - 27$. Comparing this with the given options, we see that option 1 is the correct answer.
### Examples
Polynomial multiplication, like in this problem, is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and stress on different parts of the structure. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces for 3D models.
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