Problem 22 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ \left(x+y^{2}\right)^{3} $$

Step-by-Step Solution

Answer

The expanded form is \(x^3 + 3x^2y^2 + 3xy^4 + y^6\).

Step 1: Understand 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. In this expression, \(a = x\), \(b = y^2\), and \(n = 3\).

Step 2: Determine the Binomial Coefficients

The binomial coefficients for \(n = 3\) are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\). You can find these using Pascal's triangle or the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Step 3: Expand the Expression

Substitute the values of \(a\), \(b\), and the binomial coefficients into the theorem.\[(x+y^2)^3 = \binom{3}{0}x^3(y^2)^0 + \binom{3}{1}x^2(y^2)^1 + \binom{3}{2}x^1(y^2)^2 + \binom{3}{3}x^0(y^2)^3\]Simplify each term using the coefficients calculated earlier.

Step 4: Simplify Each Term

Compute each term individually:- \(\binom{3}{0}x^3(y^2)^0 = 1 \cdot x^3 \cdot 1 = x^3\)- \(\binom{3}{1}x^2(y^2)^1 = 3 \cdot x^2 \cdot y^2 = 3x^2y^2\)- \(\binom{3}{2}x^1(y^2)^2 = 3 \cdot x \cdot (y^2)^2 = 3xy^4\)- \(\binom{3}{3}x^0(y^2)^3 = 1 \cdot 1 \cdot y^6 = y^6\)

Step 5: Write the Final Expanded Form

Combine all simplified terms to write the final expanded expression:\[x^3 + 3x^2y^2 + 3xy^4 + y^6\]

Key Concepts

Binomial Coefficients

Binomial coefficients are the numbers that arise in the expansion of a binomial expression. These coefficients can be found using the formula \(inom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n\) is the power the binomial is raised to, and \(k\) is the term number in the expansion.

• The factorial symbol \(!\) means to multiply a series of descending natural numbers. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• In the expression \((x + y^2)^3\), the coefficients for each term were found as \(1, 3, 3, 1\), using this formula.
• These coefficients can also be visualized using Pascal's Triangle for a quick look-up.

Understanding these coefficients is critical as they help in determining the magnitude of each term when the binomial is expanded. They act as multipliers for each term in the polynomial expansion.

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial \((a + b)^n\) as a sum of terms using the binomial theorem. This helps when you need to simplify expressions and work with powers of binomials.

• The binomial theorem formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) provides a systematic method to obtain the expanded form.
• For \((x + y^2)^3\), substituting \(a = x\), \(b = y^2\), and \(n = 3\) into the formula gives four terms in the expanded polynomial.
• Each term is found by calculating powers of \(a\) and \(b\), then multiplying by the corresponding binomial coefficient.

Finally, simplifying each term individually and summing them gives the expanded polynomial: \(x^3 + 3x^2y^2 + 3xy^4 + y^6\). This process turns a compound power expression into a simpler sum of terms.

Pascal's Triangle

Pascal's Triangle is a triangular array where each row represents the coefficients of the expanded form of a binomial power. This is named after the French mathematician Blaise Pascal.

• Each number in the triangle is the sum of the two numbers directly above it from the previous row.
• For example, the third row of Pascal's Triangle is \(1, 3, 3, 1\), which matches the coefficients found for \((x + y^2)^3\).
• It provides an easy way to discover binomial coefficients without the need for calculations.

Using Pascal's Triangle is especially helpful when dealing with smaller powers, as it provides a quick reference for coefficients. This graphical tool eases the process of binomial expansion.