Question

the variables a, b, and c represent polynomials where $a = x^2$, $b = 3x + 2$, and $c = x - 3$. what is $ab - c^2$ in simplest form?
○ $3x^3 + 2x^2 - x + 3$
○ $3x^3 + 2x^2 - x - 3$
○ $3x^3 + x^2 - 6x + 9$
○ $3x^3 + x^2 + 6x - 9$

Explanation:

Step1: Calculate \( AB \)

Given \( A = x^2 \) and \( B = 3x + 2 \), we multiply them:
\( AB = x^2(3x + 2) = 3x^3 + 2x^2 \) (using the distributive property \( a(b + c)=ab + ac \)).

Step2: Calculate \( C^2 \)

Given \( C = x - 3 \), we square it:
\( C^2=(x - 3)^2 = x^2 - 6x + 9 \) (using the formula \( (a - b)^2=a^2 - 2ab + b^2 \), where \( a = x \), \( b = 3 \)).

Step3: Calculate \( AB - C^2 \)

Substitute the results from Step1 and Step2:
\( AB - C^2=(3x^3 + 2x^2)-(x^2 - 6x + 9) \)
Distribute the negative sign:
\( = 3x^3 + 2x^2 - x^2 + 6x - 9 \)
Combine like terms:
\( = 3x^3 + (2x^2 - x^2)+ 6x - 9 = 3x^3 + x^2 + 6x - 9 \).

Answer:

\( 3x^3 + x^2 + 6x - 9 \) (the fourth option)