Factor the polynomial expression for the volume of a box.
Check the final answer first, then review the worked steps.
Problem
Factor the polynomial expression for the volume of a box.
Answer
\(3(x-6)(x-6)\)
Step-by-step solution
- Identify the polynomial: The problem states that the volume of the resulting box is given by the polynomial $V = 3x^2 - 36x + 108$.
- Factor out the greatest common factor (GCF): Observe that all the coefficients (3, -36, and 108) are divisible by 3. Factoring out 3, we get $V = 3(x^2 - 12x + 36)$.
- Factor the quadratic expression: Now, we need to factor the quadratic expression inside the parentheses, which is $x^2 - 12x + 36$. This is a perfect square trinomial of the form $a^2 - 2ab + b^2 = (a-b)^2$. In this case, $a=x$ and $b=6$, since $x^2 = x^2$, $6^2 = 36$, and $2ab = 2(x)(6) = 12x$. Therefore, $x^2 - 12x + 36$ factors into $(x-6)(x-6)$ or $(x-6)^2$.
- Write the fully factored polynomial: Combining the GCF with the factored quadratic, the complete factored form of the volume polynomial is $V = 3(x-6)(x-6)$ or $V = 3(x-6)^2$.