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8. Perform the operations:

[tex]\[
(x+3)\left(x^2-2x+5\right)
\][/tex]

A. [tex]\( x^3 + x^2 - x + 15 \)[/tex]

B. [tex]\( x^4 + 15 \)[/tex]

C. [tex]\((x+3)^2 - 2(x+3) + 5x + 3\)[/tex]

9. Choose the best answer.

Sagot :

GinnyAnswer
Sure, let's go through the process of performing the operations for each given expression.

### Expression 1: [tex]\((x+3)(x^2 - 2x + 5)\)[/tex]

To expand [tex]\((x+3)(x^2 - 2x + 5)\)[/tex], we'll use the distributive property:
[tex]\[ (x + 3)(x^2 - 2x + 5) = x(x^2 - 2x + 5) + 3(x^2 - 2x + 5) \][/tex]
[tex]\[ = x^3 - 2x^2 + 5x + 3x^2 - 6x + 15 \][/tex]
Combine like terms:
[tex]\[ = x^3 + (3x^2 - 2x^2) + (5x - 6x) + 15 \][/tex]
[tex]\[ = x^3 + x^2 - x + 15 \][/tex]

So, the expanded form of [tex]\((x + 3)(x^2 - 2x + 5)\)[/tex] is:
[tex]\[ x^3 + x^2 - x + 15 \][/tex]

### Expression 2: [tex]\(x^3 + x^2 - x + 15\)[/tex]

This expression is already simplified and doesn't require any further operations. It remains:
[tex]\[ x^3 + x^2 - x + 15 \][/tex]

### Expression 3: [tex]\(x^4 + 15\)[/tex]

This expression is also already in its simplest form:
[tex]\[ x^4 + 15 \][/tex]

### Expression 4: [tex]\((x + 3)^2 - 2(x + 3) + 5x + 3\)[/tex]

First, expand [tex]\((x + 3)^2\)[/tex]:
[tex]\[ (x + 3)^2 = (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 \][/tex]

Now, distribute [tex]\(-2\)[/tex] to [tex]\((x+3)\)[/tex]:
[tex]\[ -2(x + 3) = -2x - 6 \][/tex]

Now, put all the pieces together:
[tex]\[ (x + 3)^2 - 2(x + 3) + 5x + 3 \to x^2 + 6x + 9 - 2x - 6 + 5x + 3 \][/tex]

Combine like terms:
[tex]\[ x^2 + (6x - 2x + 5x) + (9 - 6 + 3) \][/tex]
[tex]\[ = x^2 + 9x + 6 \][/tex]

So, the simplified form of [tex]\((x + 3)^2 - 2(x + 3) + 5x + 3\)[/tex] is:
[tex]\[ x^2 + 9x + 6 \][/tex]

To summarize, the results of the operations are:

1. [tex]\((x + 3)(x^2 - 2x + 5) = x^3 + x^2 - x + 15\)[/tex]
2. [tex]\(x^3 + x^2 - x + 15\)[/tex] (already simplified)
3. [tex]\(x^4 + 15\)[/tex] (already simplified)
4. [tex]\((x + 3)^2 - 2(x + 3) + 5x + 3 = x^2 + 9x + 6\)[/tex]
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