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Answer :
To find the product of the polynomials [tex]\((x^2 + 3x + 9)\)[/tex] and [tex]\((x - 3)\)[/tex], let's carry out the multiplication step-by-step.
Given:
[tex]\[
P(x) = (x^2 + 3x + 9) \cdot (x - 3)
\][/tex]
We'll use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials.
First, distribute [tex]\(x\)[/tex] from [tex]\((x - 3)\)[/tex] to each term in [tex]\((x^2 + 3x + 9)\)[/tex]:
[tex]\[
x \cdot (x^2 + 3x + 9) = x(x^2) + x(3x) + x(9) = x^3 + 3x^2 + 9x
\][/tex]
Next, distribute [tex]\(-3\)[/tex] from [tex]\((x - 3)\)[/tex] to each term in [tex]\((x^2 + 3x + 9)\)[/tex]:
[tex]\[
-3 \cdot (x^2 + 3x + 9) = -3(x^2) + (-3)(3x) + (-3)(9) = -3x^2 - 9x - 27
\][/tex]
Now, combine the results of the two distributions:
[tex]\[
(x^3 + 3x^2 + 9x) + (-3x^2 - 9x - 27)
\][/tex]
Combine like terms:
[tex]\[
x^3 + (3x^2 - 3x^2) + (9x - 9x) - 27 = x^3 - 27
\][/tex]
So, the product of [tex]\((x^2 + 3x + 9)\)[/tex] and [tex]\((x - 3)\)[/tex] is:
[tex]\[
x^3 - 27
\][/tex]
Given the choices:
1. [tex]\( x^3 - 27 \)[/tex]
2. [tex]\( x^2 + 4x + 6 \)[/tex]
3. [tex]\( x^3 - 6x^2 - 18x - 27 \)[/tex]
4. [tex]\( -6x^4 + x^3 - 18x^2 - 27 \)[/tex]
The correct answer is:
[tex]\[
(1) \quad x^3 - 27
\][/tex]
Given:
[tex]\[
P(x) = (x^2 + 3x + 9) \cdot (x - 3)
\][/tex]
We'll use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials.
First, distribute [tex]\(x\)[/tex] from [tex]\((x - 3)\)[/tex] to each term in [tex]\((x^2 + 3x + 9)\)[/tex]:
[tex]\[
x \cdot (x^2 + 3x + 9) = x(x^2) + x(3x) + x(9) = x^3 + 3x^2 + 9x
\][/tex]
Next, distribute [tex]\(-3\)[/tex] from [tex]\((x - 3)\)[/tex] to each term in [tex]\((x^2 + 3x + 9)\)[/tex]:
[tex]\[
-3 \cdot (x^2 + 3x + 9) = -3(x^2) + (-3)(3x) + (-3)(9) = -3x^2 - 9x - 27
\][/tex]
Now, combine the results of the two distributions:
[tex]\[
(x^3 + 3x^2 + 9x) + (-3x^2 - 9x - 27)
\][/tex]
Combine like terms:
[tex]\[
x^3 + (3x^2 - 3x^2) + (9x - 9x) - 27 = x^3 - 27
\][/tex]
So, the product of [tex]\((x^2 + 3x + 9)\)[/tex] and [tex]\((x - 3)\)[/tex] is:
[tex]\[
x^3 - 27
\][/tex]
Given the choices:
1. [tex]\( x^3 - 27 \)[/tex]
2. [tex]\( x^2 + 4x + 6 \)[/tex]
3. [tex]\( x^3 - 6x^2 - 18x - 27 \)[/tex]
4. [tex]\( -6x^4 + x^3 - 18x^2 - 27 \)[/tex]
The correct answer is:
[tex]\[
(1) \quad x^3 - 27
\][/tex]
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