Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \((3 \sqrt{x}+5)^{3}\)

Step-by-Step Solution

Answer

The expanded and simplified form of the expression \((3 \sqrt{x}+5)^{3}\) using the Binomial Theorem is \( 27x^{3/2} + 405x + 675\sqrt{x} + 125 \)

Step 1: Identify the values of n, a, and b in the Binomial Theorem

In the expression \((3 \sqrt{x}+5)^{3}\), \( n = 3 \), \( a = 3 \sqrt{x} \), and \( b = 5 \).

Step 2: Substitute the values into the Binomial Theorem

Substitute these values into the Binomial Theorem \( (a+b)^{n} = a^{n} + {n \choose 1}a^{n-1}b + {n \choose 2}a^{n-2}b^{2} + ... + b^{n} \), to get: \( (3 \sqrt{x}+5)^{3} = (3 \sqrt{x})^{3} + {3 \choose 1}(3 \sqrt{x})^{2}(5) + {3 \choose 2}(3 \sqrt{x})(5)^{2} + (5)^{3} \).

Step 3: Evaluate binomial coefficients and simplify

Evaluate the binomial coefficients and simplify the terms: \( (3 \sqrt{x})^{3} = 27x^{3/2}, {3 \choose 1} = 3, (3 \sqrt{x})^{2}(5) = 135x, {3 \choose 2} = 3, (3 \sqrt{x})(5)^{2} = 225 \sqrt{x}, (5)^{3} = 125. \nThe simplified equation becomes: \( 27x^{3/2} + 3*135x + 3*225\sqrt{x} + 125 \)

Step 4: Further simplify the equation

Further simplify to get the expanded expression: \( 27x^{3/2} + 405x + 675\sqrt{x} + 125 \)

Key Concepts

Binomial Expansion

Binomial expansion is a key concept in algebra that allows us to express expressions of the form \((a+b)^n\) in a longer polynomial form. The Binomial Theorem provides a systematic way of doing this expansion.
This theorem states that for any positive integer \(n\), the expression \((a+b)^n\) can be expanded and written as a sum of terms involving coefficients known as binomial coefficients.
These coefficients are derived using the combination formula. For example, in expanding \((3 \sqrt{x}+5)^3\):

• Identify the terms \(a = 3\sqrt{x}\) and \(b = 5\).
• Use the Binomial Theorem formula: \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\).
• This results in the expression \((3 \sqrt{x}+5)^3 = (3 \sqrt{x})^3 + {3 \choose 1}(3 \sqrt{x})^2(5) + {3 \choose 2}(3 \sqrt{x})(5)^2 + (5)^3\).

Algebra Simplification

Algebra simplification involves breaking down complex expressions into simpler, more manageable ones. After applying the Binomial Theorem, the next step is simplifying each term separately.
Simplifying requires the careful combination and reduction of terms using arithmetic and algebraic rules.
For instance, in the expression we expanded earlier:

• Calculate each term separately such as \((3 \sqrt{x})^3 = 27x^{3/2}\).
• Apply coefficients and simplify multiplication, like \({3 \choose 1} \times 135x\).
• Sum up all terms post-simplification to achieve the neatest polynomial form, \(27x^{3/2} + 405x + 675\sqrt{x} + 125\).

Through this process, the expression is transformed into something easier to work with or interpret in further mathematical operations.

Exponentiation

Exponentiation is a mathematical operation involving numbers raised to powers, which is a crucial aspect in expressions like \((3 \sqrt{x} + 5)^3\). In such polynomials, exponentiation is applied initially to each term \(a\) and \(b\) separately.
This process requires strong knowledge of rules governing exponents, including that multiplying like bases results in the exponents being added. For example:

• Raise \((3 \sqrt{x})\) to the power of 3, which results in \(27x^{3/2}\), using the laws of exponents (\(a^m \times a^n = a^{m+n}\)).
• Recognize the distinct addition rules while handling \(5^3 = 125\).

Understanding exponentiation allows for accurate expansion and simplification within the binomial expression.

Mathematical Coefficients

Mathematical coefficients are the numerical factors in polynomial terms. In binomial expansions, these coefficients are crucial in determining the weight each term has in the expanded structure.
These are derived based on the Binomial Theorem’s combination formula: \({n \choose k} = \frac{n!}{k!(n-k)!}\).

• The coefficients show up in the expanded terms such as \({3 \choose 1}\) and \({3 \choose 2}\), which resolve to specific numerical values like 3.
• These coefficients are multiplied with the results of exponentiated terms like \((3 \sqrt{x})^{2} \times (5)\).

Accurate calculation of coefficients ensures each term in the polynomial reflects the correct mathematical relationships defined in the original expression.