Problem 101 - Algebra 2

Question

Expand each binomial. $$ (x+4)^{8} $$

Step-by-Step Solution

Answer

The expanded form of \( (x+4)^8 \) is \(x^8 + 32x^7 + 448x^6 + 3584x^5 + 17920x^4 + 57344x^3 + 114688x^2 + 147456x + 65536\). Using the binomial theorem, we can systematically expand and simplify the polynomial.

Step 1: Identify the components of the binomial

The components of the binomial are \(x\) and \(4\), and the power of the binomial is \(8\).

Step 2: Use Binomial theorem

The Binomial Theorem states that: \((a+b)^n= \sum_{k=0}^{n} C(n,k) * a^{n-k} * b^k\) where \(C(n, k)\) is the binomial coefficient calculated as \(C(n, k) = n! / [k!(n-k)!]\). In our case, \(a = x\), \(b = 4\), and \(n = 8\). We apply this formula to each term of our expansion.

Step 3: Calculate binomial coefficients and simplify terms

We apply the binomial theorem to all terms from \(k = 0\) to \(k = 8\), calculating binomial coefficients, raising \(x\) and \(4\) to respective powers, and simplifying each term by multiplying them together. Do this for all 9 terms and add them together to get the final answer.

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows the expansion of expressions in the form of \((a + b)^n\). By using this theorem, we can easily expand these expressions into a sum involving terms of the original components raised to different powers.
The formula for the binomial theorem is expressed as:

• \((a + b)^n = \sum_{k=0}^{n} C(n,k) \cdot a^{n-k} \cdot b^k\)

Here, \(a\) and \(b\) are the terms of the binomial expression, \(n\) is the power to which the binomial is raised, and \(C(n,k)\) is the binomial coefficient corresponding to the term. The expansion provides \(n+1\) terms, each consisting of a combination of \(a\) and \(b\) raised to varying powers.
The beauty of the binomial theorem lies in its ability to simplify complex polynomial expressions, making it an essential concept in algebra and beyond.

Binomial Coefficient

Binomial coefficients are fundamental components of the binomial theorem, providing the specific weights for each term in the expansion. These coefficients are represented by \(C(n,k)\) and are mathematically defined as:

• \(C(n,k) = \frac{n!}{k!(n-k)!}\)

Where \(n!\) denotes \(n\) factorial, which is the product of all positive integers from 1 to \(n\). Similarly, \(k!\) is \(k\) factorial, and \((n-k)!\) represents the factorial of \(n-k\).
In practice, to find a specific binomial coefficient, you simply substitute the relevant values into this formula. For example, to get a term involving \(x^5\) in the expansion of \((x+4)^8\), you would calculate \(C(8,3)\) to combine it with the powers of \(x\) and \(4\).
Understanding how to calculate binomial coefficients enables you to construct polynomial expansions effectively, making them easier to manage and solve.

Polynomial Expansion

Polynomial expansion involves breaking down expressions raised to a power into a sum of individual terms, which can be easily managed and analyzed. In the case of expanding \((x+4)^8\), the polynomial expansion transforms the binomial expression into a sum of terms involving powers of \(x\) and 4.
Each term in the expanded form is derived from the general term \(C(n,k) \cdot a^{n-k} \cdot b^k\) in the binomial theorem. This means for each increase in \(k\), the power of \(x\) decreases while the power of 4 increases.

• The first term is the coefficient times \(x^8\), then \(8x^7 \cdot 4\), continuing this pattern until \(4^8\).

The complete polynomial expansion reveals all possible combinations of \(x\) and \(4\) raised to appropriate powers and combined with their respective coefficients.
This systematic approach to finding expansion makes understanding algebraic expressions simpler and more structured.

Algebra 2 Concepts

In Algebra 2, understanding polynomial expressions and their expansions is a key skill. Binomial expansion is just one of many algebraic techniques that helps to simplify and solve polynomial equations. It leverages various foundational concepts such as exponents, factorials, and series expansion.
In this particular context, Algebra 2 emphasizes:

• Recognizing and applying the binomial theorem to expand expressions.
• Calculating and understanding the significance of binomial coefficients.
• Using expansions to simplify algebraic expressions and facilitate easier computation.

These concepts are not only crucial for higher-level mathematics but also for their practical applications in real-world problems and advanced mathematical theories.
Mastery of Algebra 2 concepts such as binomial expansion ensures a solid foundation for further mathematical learning and real-life problem-solving.