Problem 36 - Contemporary Precalculus

Question

Expand and (where possible) simplify the expression. $$(\sqrt{3}+1)^{6}-(\sqrt{3}-1)^{6}$$

Step-by-Step Solution

Answer

$$ Answer: The simplified expression is $$2080$$.

Step 1: Recall the binomial theorem

The binomial theorem states that $$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k,$$ where \(\binom{n}{k}\) is the binomial coefficient, which can be computed as $$\binom{n}{k}=\frac{n!}{k!(n-k)!}.$$

Step 2: Expand the first term using the binomial theorem

Using the binomial theorem, we expand the first term as follows: $$(\sqrt{3}+1)^6=\sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k}(1)^k.$$

Step 3: Expand the second term using the binomial theorem

Similarly, we expand the second term as follows: $$(\sqrt{3}-1)^6=\sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k}(-1)^k.$$

Step 4: Subtract the two expanded terms

To subtract the expanded expressions, we combine like terms, taking into account the signs of the coefficients: $$\sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k}(1)^k - \sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k}(-1)^k.$$

Step 5: Simplify the difference

Combine the sums and simplify the expression: \begin{align*} & \sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k}((1)^k-(-1)^k) \\ &= \binom{6}{0}(\sqrt{3})^6+\binom{6}{2}(\sqrt{3})^4+\binom{6}{4}(\sqrt{3})^2+\binom{6}{6} \\ &= 1\cdot 729 + 15\cdot 81 + 15\cdot 9 + 1 \\ &= 729 + 1215 + 135 + 1 \\ &= 2080. \end{align*} So, the simplified expression is as follows: $$(\sqrt{3}+1)^{6}-(\sqrt{3}-1)^{6} = 2080.$$

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra, involving the transformation of a compact expression raised to a power into a sum of terms. When expanding a polynomial like \((a + b)^n\), the terms are arranged according to the powers of \(a\) and \(b\). Each term in this expansion is called a binomial term. To expand, you increase the power of \(b\) while decreasing that of \(a\) consistently. This process continues until you've achieved all combinations for the given power.

For instance, in the exercise, the expression \((\sqrt{3} + 1)^6\) is expanded into individual terms featuring decreasing powers of \(\sqrt{3}\) and increasing powers of \(1\). Similarly, the same process applies to \((\sqrt{3} - 1)^6\), where \(-1\) is used instead of \(1\). Each expansion generates terms that are crucial for later simplification.

Using polynomial expansion allows us to work with complex expressions by breaking them into manageable parts.

Algebraic Simplification

Algebraic simplification involves reducing an expanded expression to its simplest form. This can involve combining like terms, factoring, or applying other algebraic principles to make the expression more concise. In the given exercise, after expanding both \((\sqrt{3} + 1)^6\) and \((\sqrt{3} - 1)^6\), the next crucial step is simplification.

The expanded forms are subtracted to yield a new expression.

• Each like term, such as those with same powers of \(\sqrt{3}\), is combined.
• Further, the operation \((1)^k - (-1)^k\) leads to some terms canceling out based on the parity of \(k\).

This process ultimately transforms the expression into some key remaining terms that add up to form the simplified result of 2080. Simplifying expressions makes complex mathematics more approachable and easier to interpret.

Binomial Coefficients

Binomial coefficients are central to the binomial theorem and polynomial expansion, denoting the number of ways to choose elements from a set. These coefficients, represented as \(\binom{n}{k}\), are computed using factorials. The formula is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). These components describe the weight or coefficient of each term in a binomial expansion.

In our given problem, the binomial coefficients determine the multiplicative factor for each term in the expansions of \((\sqrt{3} + 1)^6\) and \((\sqrt{3} - 1)^6\).

• For example, \(\binom{6}{2}\) determines how many ways two items can be selected from six items, influencing the coefficient of the \((\sqrt{3})^{4}\) term.
• This coefficient system in connection with powers of expansion allows for precise expression manipulation.

Thus, understanding binomial coefficients plays a pivotal role in algebraic procedures related to binomial expansion and is the backbone of calculating terms in polynomials.