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The variables [tex]\( A, B \)[/tex], and [tex]\( C \)[/tex] represent polynomials where [tex]\( A = x^2 \)[/tex], [tex]\( B = 3x + 2 \)[/tex], and [tex]\( C = x - 3 \)[/tex]. What is [tex]\( AB - C^2 \)[/tex] in simplest form?

A. [tex]\( 3x^3 + 2x^2 - x + 3 \)[/tex]
B. [tex]\( 3x^3 + 2x^2 - x - 3 \)[/tex]
C. [tex]\( 3x^3 + x^2 - 6x + 9 \)[/tex]
D. [tex]\( 3x^3 + x^2 + 6x - 9 \)[/tex]



Answer :

GinnyAnswer
Let's break down the problem step-by-step given the polynomials [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex]:

1. Define the polynomials:
[tex]\[ A = x^2 \][/tex]
[tex]\[ B = 3x + 2 \][/tex]
[tex]\[ C = x - 3 \][/tex]

2. Compute [tex]\(AB\)[/tex]:
[tex]\[ AB = A \cdot B = x^2 \cdot (3x + 2) \][/tex]
We distribute [tex]\(x^2\)[/tex] across each term inside the parentheses:
[tex]\[ AB = x^2 \cdot 3x + x^2 \cdot 2 = 3x^3 + 2x^2 \][/tex]

3. Compute [tex]\(C^2\)[/tex]:
[tex]\[ C^2 = (x - 3)^2 \][/tex]
We use the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[ C^2 = x^2 - 2 \cdot x \cdot 3 + 3^2 = x^2 - 6x + 9 \][/tex]

4. Subtract [tex]\(C^2\)[/tex] from [tex]\(AB\)[/tex]:
[tex]\[ AB - C^2 = (3x^3 + 2x^2) - (x^2 - 6x + 9) \][/tex]
Distribute the negative sign and combine like terms:
[tex]\[ AB - C^2 = 3x^3 + 2x^2 - x^2 + 6x - 9 \][/tex]
Simplify:
[tex]\[ AB - C^2 = 3x^3 + (2x^2 - x^2) + 6x - 9 = 3x^3 + x^2 + 6x - 9 \][/tex]

So, the expression [tex]\(AB - C^2\)[/tex] in simplest form is:
[tex]\[ 3x^3 + x^2 + 6x - 9 \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{3 x^3 + x^2 + 6 x - 9} \][/tex]

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