Problem 52 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x+2)^{3} $$

Step-by-Step Solution

Answer

The product of the given equation \((x+2)^{3}\) is \(x^3 + 6x^2 + 12x + 8\)

Step 1: Use the Binomial theorem

According to the binomial theorem, \((x+y)^{n} = \(_{n}C_{0}*x^n*y^0 + _{n}C_{1}*x^(n-1)*y^1 + _{n}C_{2}*x^(n-2)*y^2 +...+ _{n}C_{n}*x^0*y^n\)\n\nApplying this theorem to \((x+2)^{3}\), x is replaced with x, y is replaced with 2, and n is replaced with 3.

Step 2: Expand using the binomial theorem

After replacing, we get \(_{3}C_{0}*x^3*2^0 + _{3}C_{1}*x^2*2^1 + _{3}C_{2}*x^1*2^2 + _{3}C_{3}*x^0*2^3\) which simplifies to:\n\n\(x^3 + 3x^2*2 + 3x*2^2 + 8\)

Step 3: Simplify

The final step is to simplify the expression obtained from step 2. This gives the final solution as: \(x^3 + 6x^2 + 12x + 8\)

Key Concepts

Polynomial Expansion

Polynomial expansion is a method that allows us to express a polynomial raised to a power in an expanded form. The Binomial Theorem is crucial in this process. It is particularly useful for expanding expressions of the form \((x + y)^n\). In our exercise, we see the polynomial \((x + 2)^3\). We can expand this using the Binomial Theorem, which provides a way to calculate the coefficients of each term in the expansion through combinations and powers of \(x\) and \(y\).
By following the binomial theorem formula, each term in the expansion is constructed by taking the sum of certain multiplicative combinations involving the variables and constants from the original polynomial. This helps in identifying each individual term within the expanded polynomial in a systematic approach.

Combination Formula

The Combination Formula is an essential part of expanding a polynomial using the Binomial Theorem. This formula helps determine how coefficients are calculated. The combination number is represented as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\),where \(n\) is the total number of items to pick from, \(k\) is the number of items to choose, and \(!\) denotes a factorial.
In our example of \((x+2)^3\), the combination numbers are used to assign the coefficient to each term in the expanded form. Specifically, for change \((n,k)\) to values like \((3,0), (3,1), (3,2),\) and \( (3,3)\), we find the coefficients (\(1, 3, 3,\) and \(1\)) for respective terms.
These combination numbers allow us to quickly determine the multiplicative impact of each term within the polynomial expansion, making the process much more efficient and manageable.

Algebraic Simplification

Once a polynomial is expanded using the Binomial Theorem and combination formula, the next step is algebraic simplification. This involves reducing the expression to its simplest form by performing operations like multiplication and addition to merge terms.
In the specific example of \(x^3 + 3x^2 \cdot 2 + 3x \cdot 2^2 + 8\),we need to simplify each term:

• \(x^3\) remains unchanged as no further simplification is needed.
• \(3x^2 \cdot 2\) simplifies to \(6x^2\) by multiplying the numbers.
• \(3x \cdot 2^2\) simplifies to \(12x\), again by applying multiplication.
• \(2^3 = 8\) remains \(8\) as it is a constant term.

Combining all these simplified terms together yields the final result: \(x^3 + 6x^2 + 12x + 8\). This simplification stage is crucial for achieving a clean, usable form of the polynomial that can be easily interpreted or used in further calculations.