Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((2 x+3 y)^{5}\)

Step-by-Step Solution

Answer

Expanded form of \((2x+3y)^5\) is: \(32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5\).

Step 1 - Determine the coefficients

The coefficients of the expansion are determined by the 5th row of Pascal's triangle (1, 5, 10, 10, 5, 1). These numbers will be multiplied with the terms in the following steps.

Step 2 - Apply powers to the first term of the binomial

Start from the highest power of 5 and go down to 0. The powers of the first term (2x) in descending order are: \((2x)^5\), \((2x)^4\), \((2x)^3\), \((2x)^2\), \(2x^1\), \(2x^0\) respectively.

Step 3 - Apply powers to the second term of the binomial

Start from the lowest power of 0 and go upto 5. The powers of the second term (3y) in ascending order are: \((3y)^0\), \((3y)^1\), \((3y)^2\), \((3y)^3\), \((3y)^4\), \((3y)^5\) respectively.

Step 4 - Combine coefficients with the terms and simplify

Combine each corresponding pair from steps 2 and 3 with the coefficients from step 1: \(1*(2x)^5*(3y)^0, 5*(2x)^4*(3y)^1, 10*(2x)^3*(3y)^2, 10*(2x)^2*(3y)^3, 5*(2x)^1*(3y)^4, 1*(2x)^0*(3y)^5\). Now you can simplify each term and add them together to get the final expanded form.

Key Concepts

Binomial Expansion

Binomial expansion is a way to express a binomial raised to a power as a sum of terms. A binomial is an algebraic expression that involves exactly two distinct terms, such as \((2x + 3y)\). When you expand a binomial, you are rewriting it in an expanded form where each term contains both parts of the binomial raised to varying powers.
The expansion process for a binomial raised to a power involves several steps:

• Identify the Binomial: Determine the two terms in the expression. For instance, in \((2x + 3y)\), the terms are \(2x\) and \(3y\).
• Determine the Power: Note the power to which the binomial is raised, such as \(5\) in \((2x + 3y)^5\).
• Use Pascal's Triangle: Find the row of Pascal's Triangle that corresponds to the power. The coefficients from the triangle will be used in forming the expanded terms.
• Combine Terms: Apply the coefficients and the powers of each term as you expand. Each term in the expansion results from varying combinations of the powers of the two components in the binomial.

This method of expanding binomials makes use of both binomial coefficients and the respective powers of each part of the expression to give a series of terms that collectively equal the original binomial expression raised to a specified power.

Coefficients

In binomial expansion, coefficients play a crucial role in shaping the expansion's outcome. Coefficients are the numeric factors that multiply each term in the expanded form of the binomial. To find these coefficients, we often use Pascal's Triangle, a special array of numbers.
Pascal's Triangle is organized in a triangular format where each row represents the coefficients for the powers of a binomial expansion. For instance, the fifth row of Pascal's Triangle, which is \(1, 5, 10, 10, 5, 1\), provides the coefficients for \((x + y)^5\).

• First Coefficient: The first term in a binomial expansion always has a coefficient of \(1\).
• Middle Coefficients: These numbers increase to a peak in the middle of Pascal's Triangle and symmetrically decrease.
• Last Coefficient: The final coefficient also equals \(1\), mirroring the first.

The coefficients multiply the corresponding terms derived from the respective powers of each binomial component. This multiplication adjusts the contribution of each binomial term in the final expanded expression.

Powers of Binomials

When expanding a binomial expression, it is important to take into account the powers of each term in the binomial. Binomials like \((2x + 3y)^n\) involve raising both terms to various powers and combining them according to rules of binomial expansion.
In the expansion process, each term from the binomial is raised to a particular power that changes systematically across each term in the series:

• Descending Powers of the First Term: Starting from the highest power \(n\), the first term of the binomial (here \(2x\)) is raised to decreasing powers until it reaches zero. So, for \(n = 5\), it would go as \((2x)^5, (2x)^4, ..., (2x)^0\).
• Ascending Powers of the Second Term: Conversely, the second term (\(3y\)) starts at a power of zero, increasing up to \(n\). For \(n = 5\), the powers go as \((3y)^0, (3y)^1, ..., (3y)^5\).

By combining the powers of both terms with the appropriate coefficients from Pascal's Triangle, an accurate expanded expression is formed that faithfully represents the original binomial raised to its stated power. This expansion captures all possible products of the two terms raised to their respective powers within the set sequence.