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The variables [tex]$A, B$[/tex], and [tex]$C$[/tex] represent polynomials where [tex]$A=x+1$[/tex], [tex]$B=x^2+2x-1$[/tex], and [tex]$C=2x$[/tex]. What is [tex]$AB + C$[/tex] in simplest form?

A. [tex]$x^3 + 3x - 1$[/tex]
B. [tex]$x^3 + 4x - 1$[/tex]
C. [tex]$x^3 + 3x^2 + 3x - 1$[/tex]
D. [tex]$x^3 + 2x^2 - x + 1$[/tex]



Answer :

GinnyAnswer
Let's find the expression for [tex]\(A \cdot B + C\)[/tex] given the polynomials [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:

1. Define the polynomials:
- [tex]\(A = x + 1\)[/tex]
- [tex]\(B = x^2 + 2x - 1\)[/tex]
- [tex]\(C = 2x\)[/tex]

2. Calculate [tex]\(A \cdot B\)[/tex]:
[tex]\[ A \cdot B = (x + 1)(x^2 + 2x - 1) \][/tex]
Use the distributive property to expand this product:
[tex]\[ (x + 1)(x^2 + 2x - 1) = x \cdot (x^2 + 2x - 1) + 1 \cdot (x^2 + 2x - 1) \][/tex]
Compute each term:
[tex]\[ x \cdot (x^2 + 2x - 1) = x^3 + 2x^2 - x \][/tex]
[tex]\[ 1 \cdot (x^2 + 2x - 1) = x^2 + 2x - 1 \][/tex]
Combine these results:
[tex]\[ x^3 + 2x^2 - x + x^2 + 2x - 1 = x^3 + 3x^2 + x - 1 \][/tex]

3. Add [tex]\(C\)[/tex] to the result:
[tex]\[ A \cdot B + C = (x^3 + 3x^2 + x - 1) + 2x \][/tex]
Combine like terms:
[tex]\[ x^3 + 3x^2 + x + 2x - 1 = x^3 + 3x^2 + 3x - 1 \][/tex]

Thus, the simplest form of [tex]\(A \cdot B + C\)[/tex] is:
[tex]\[ \boxed{x^3 + 3x^2 + 3x - 1} \][/tex]

This matches one of the provided choices:
- [tex]\(x^3 + 3 x - 1\)[/tex]
- [tex]\(x^3 + 4 x - 1\)[/tex]
- [tex]\(x^3 + 3 x^2 + 3 x - 1\)[/tex]
- [tex]\(x^3 + 2 x^2 - x + 1\)[/tex]

So, the correct answer is:
[tex]\[ x^3 + 3 x^2 + 3 x - 1 \][/tex]

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