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[tex]$A$[/tex], [tex]$B$[/tex], and [tex]$C$[/tex] are polynomials, where:

[tex]\[ \begin{array}{l}
A = 3x - 4 \\
B = x + 7 \\
C = x^2 + 2
\end{array} \][/tex]

What is [tex]\( A^2 - (B + C) \)[/tex] in simplest form?

A. [tex]\( 8x^2 - 25x + 7 \)[/tex]
B. [tex]\( 8x^2 - 25x + 11 \)[/tex]
C. [tex]\( 10x^2 - 25x + 7 \)[/tex]
D. [tex]\( 10x^2 - 25x + 11 \)[/tex]

Sagot :

GinnyAnswer
Let's simplify the expression [tex]\( A^2 - (B + C) \)[/tex] step by step, given [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] as follows:

[tex]\[ A = 3x - 4 \][/tex]
[tex]\[ B = x + 7 \][/tex]
[tex]\[ C = x^2 + 2 \][/tex]

1. Calculate [tex]\( A^2 \)[/tex]:
[tex]\[ A^2 = (3x - 4)^2 \][/tex]
[tex]\[ = (3x - 4)(3x - 4) \][/tex]
[tex]\[ = 9x^2 - 12x - 12x + 16 \][/tex]
[tex]\[ = 9x^2 - 24x + 16 \][/tex]

2. Calculate [tex]\( B + C \)[/tex]:
[tex]\[ B + C = (x + 7) + (x^2 + 2) \][/tex]
[tex]\[ = x + 7 + x^2 + 2 \][/tex]
[tex]\[ = x^2 + x + 9 \][/tex]

3. Subtract [tex]\( B + C \)[/tex] from [tex]\( A^2 \)[/tex]:
[tex]\[ A^2 - (B + C) = 9x^2 - 24x + 16 - (x^2 + x + 9) \][/tex]
Now, distribute the negative sign:
[tex]\[ = 9x^2 - 24x + 16 - x^2 - x - 9 \][/tex]

4. Combine like terms:
[tex]\[ = (9x^2 - x^2) + (-24x - x) + (16 - 9) \][/tex]
[tex]\[ = 8x^2 - 25x + 7 \][/tex]

Therefore, the simplest form of [tex]\( A^2 - (B + C) \)[/tex] is:
[tex]\[ 8x^2 - 25x + 7 \][/tex]

So, the correct answer is:
[tex]\[ 8x^2 - 25x + 7 \][/tex]
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