Problem 73 - Algebra 2

Question

Expand each binomial. $$ (3-x)^{5} $$

Step-by-Step Solution

Answer

The expanded form of the binomial \((3-x)^{5}\) is \(243 - 405x + 270x^{2} - 90x^{3} + 15x^{4} - x^{5}\).

Step 1 - Substitution

Substitute \(a = 3\), \(b = -x\) and \(n = 5\) into the binomial theorem. This gives:\[ (3-x)^{5} = \sum_{k=0}^{5} \binom{5}{k} 3^{5-k} (-x)^{k} \]

Step 2 - Expansion

Expand the binomial and compute each term individually:\[ = \binom{5}{0} 3^{5-0} (-x)^{0} + \binom{5}{1} 3^{5-1} (-x)^{1} + \binom{5}{2} 3^{5-2} (-x)^{2} + \binom{5}{3} 3^{5-3} (-x)^{3} + \binom{5}{4} 3^{5-4} (-x)^{4} + \binom{5}{5} 3^{5-5} (-x)^{5} \= 1 \cdot 3^{5} \cdot 1 - 5 \cdot 3^{4} \cdot x + 10 \cdot 3^{3} \cdot x^{2} - 10 \cdot 3^{2} \cdot x^{3} + 5 \cdot 3 \cdot x^{4} -1 \cdot x^{5} \= 243 - 405x + 270x^{2} - 90x^{3} + 15x^{4} - x^{5}\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool. It helps us expand expressions raised to a power, like \((a + b)^n\). Instead of multiplying everything out the hard way, this theorem gives us a systematic way to do it. The formula uses the concept of combinations and involves a sum of terms.
Each term in the expansion can be calculated using a specific formula:

• \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

Here, \( \binom{n}{k} \) is a combination, explaining how many ways we can choose \(k\) items from \(n\), and it's also called a binomial coefficient. This formula is universal for all sorts of binomial expressions.
Understanding the Binomial Theorem can make seemingly complicated algebraic expressions much more manageable.

Exponentiation

Exponentiation refers to raising a number or expression to a power. It’s like multiplying the number by itself multiple times. For instance, raising 3 to the power of 5, written as \(3^5\), means multiplying 3 by itself five times: 3 × 3 × 3 × 3 × 3.
In the context of the binomial expansion, the powers of each component of the expression get distributed across the terms. This is evident in the expression \((3-x)^5\); the numbers 3 and \(-x\) are each raised to varying powers across the terms.
Each term in the expansion has a certain power of 3 and \(-x\), demonstrating how the original expression "unfolds" into multiple terms, each with its own unique power. Exponentiation is thus a core part of expanding polynomials because it dictates how each portion of a term is derived.

Polynomial Expansion

Polynomial expansion is about expressing a power of a binomial as a sum of simpler terms. This expansion helps us transform expressions like \((3-x)^5\) into a series of individual polynomial terms.

• The polynomial expansion results in terms that have decreasing powers of the first part of the binomial (like 3) and increasing powers of the second part (like \(-x\)).
• Polynomial expansion gives us a comprehensive view of how each part of the binomial contributes to the final expanded form.

For \((3-x)^5\), expansion reveals terms like 243, \(-405x\), and so on, each calculated using the Binomial Theorem and specific powers applied as per exponentiation.
Getting familiar with polynomial expansions benefits students keen on understanding complex algebraic manipulation and problem-solving.

Combination Formula

The combination formula is a central element in calculating the terms of a binomial expansion. The formula itself, shown as \( \binom{n}{k} \), is a way to determine how many ways you can choose \(k\) items from \(n\) items without considering the order.
Mathematically, it is given by:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Where \(!\) denotes factorial, meaning a product of all positive integers up to that number. This formula helps identify the coefficients that appear in the binomial expansion terms.
In the problem involving \((3-x)^5\), the combination formula helps determine coefficients like 1, 5, 10, ensuring the expansion is accurate. Recognizing how these coefficients work together with powers provides deeper insight into polynomial expressions.