Expand and simplify:
$(3+2 x)^3-(1-3 x)^2$
We need to expand and simplify the polynomial expression \((3+2x)^3 - (1-3x)^2\). To do this, we will use the algebraic identity for the cube of a binomial, \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\), and the identity for the square of a binomial, \((a-b)^2 = a^2 - 2ab + b^2\). This approach allows us to break down the problem into smaller, manageable expansions, which we will then combine to reach the final simplified form.
First, we expand the cubic term \((3+2x)^3\). Using the formula \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) with \(a=3\) and \(b=2x\), we get:
\[(3+2x)^3 = 3^3 + 3(3^2)(2x) + 3(3)(2x)^2 + (2x)^3\]
\[= 27 + 3(9)(2x) + 3(3)(4x^2) + 8x^3\]
\[= 27 + 54x + 36x^2 + 8x^3\]
Rearranging by powers of \(x\), this is \(8x^3 + 36x^2 + 54x + 27\).
Next, we expand the quadratic term \((1-3x)^2\). Using the formula \((a-b)^2 = a^2 - 2ab + b^2\) with \(a=1\) and \(b=3x\), we get:
\[(1-3x)^2 = 1^2 - 2(1)(3x) + (3x)^2\]
\[= 1 - 6x + 9x^2\]
Rearranging by powers of \(x\), this is \(9x^2 - 6x + 1\).
Now, we subtract the result of the expansion in step 3 from the result of the expansion in step 2:
\[(8x^3 + 36x^2 + 54x + 27) - (9x^2 - 6x + 1)\]
Distributing the negative sign across the second parenthesis gives:
\[8x^3 + 36x^2 + 54x + 27 - 9x^2 + 6x - 1\]
Grouping the like terms:
\[8x^3 + (36x^2 - 9x^2) + (54x + 6x) + (27 - 1)\]
\[= 8x^3 + 27x^2 + 60x + 26\]
Thus, the expanded and simplified expression is \(8x^3 + 27x^2 + 60x + 26\).
Expanding polynomials is a fundamental skill in algebra used in various fields, such as engineering and physics, to model complex systems. For instance, if you are designing a storage box where the dimensions are variables based on some parameter \(x\), the total volume might be represented by a cubic polynomial. Simplifying these expressions allows you to analyze how changing the parameter \(x\) affects the overall volume, which is essential for optimization and resource planning.