Question

the variables a, b, and c represent polynomials where a = x + 1, b = x² + 2x - 1, and c = 2x. what is ab + c in simplest form?
options:
○ ( x^3 + 3x - 1 )
○ ( x^3 + 4x - 1 )
○ ( x^3 + 3x^2 + 3x - 1 )
○ ( x^3 + 2x^2 - x + 1 )

Explanation:

Step1: Calculate \( AB \)

Given \( A = x + 1 \) and \( B = x^2 + 2x - 1 \), we use the distributive property (FOIL method for polynomials):

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Step2: Calculate \( AB + C \)

Given \( C = 2x \), we add \( C \) to the result of \( AB \):

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Wait, there is a mistake in the previous calculation. Let's re - calculate \( AB \) correctly:

Step1 (Correct): Calculate \( AB \)

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Step2 (Correct): Calculate \( AB + C \)

Now, add \( C = 2x \) to \( AB \):

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Wait, but let's check the options again. Wait, maybe I made a mistake in the problem reading. Wait, the options include \( x^3+3x^2 + 3x - 1 \)? Wait, the options given are:

1. \( x^3+3x - 1 \)
2. \( x^3+4x - 1 \)
3. \( x^3+3x^2 + 3x - 1 \)
4. \( x^3+2x^2 - x + 1 \)

Wait, let's re - do the \( AB \) calculation:

\( A=x + 1 \), \( B=x^2+2x - 1 \)

\( AB=(x)(x^2)+x(2x)+x(-1)+1(x^2)+1(2x)+1(-1) \)

\(=x^3+2x^2 - x+x^2+2x - 1 \)

\(=x^3+(2x^2+x^2)+(-x + 2x)-1 \)

\(=x^3 + 3x^2+x - 1 \)

Then \( C = 2x \), so \( AB + C=x^3 + 3x^2+x - 1+2x=x^3 + 3x^2+3x - 1 \), which is the third option.

Answer:

\( x^3 + 3x^2 + 3x - 1 \) (the third option among the given choices)