Problem 52 - Algebra 2

Question

GENETICS. Suppose \(R\) and \(W\) represent two genes that a plant can inherit from its parents. The terms of the expansion of \((R+W)^{2}\) represent the possible pairings of the genes in the offspring. Write \((R+W)^{2}\) as a polynomial.

Step-by-Step Solution

Answer

The polynomial is \( R^2 + 2RW + W^2 \).

Step 1: Identify the Expression

The problem involves the expression \((R + W)^2\), which represents the possible pairings of the genes in the offspring.

Step 2: Use the Binomial Theorem

The binomial theorem states that \((a + b)^n\) can be expanded as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). In this case, let \(a = R\) and \(b = W\), with \(n = 2\).

Step 3: Compute the Expansion

Using the binomial theorem, the expansion of \((R + W)^2\) is \( \binom{2}{0} R^2 W^0 + \binom{2}{1} R^1 W^1 + \binom{2}{2} R^0 W^2 \).

Step 4: Simplify Each Term

Calculate each term using binomial coefficients: \( \binom{2}{0} = 1 \), \( \binom{2}{1} = 2 \), and \( \binom{2}{2} = 1 \). This results in the terms: \(1R^2 + 2RW + 1W^2\).

Step 5: Write the Polynomial

Combine the simplified terms to express the polynomial as \( R^2 + 2RW + W^2 \).

Key Concepts

Polynomial Expansion

Polynomial expansion is a process of expressing a power of a binomial as a polynomial. In our example, we need to expand \((R + W)^2\), which is a binomial raised to the second power. This binomial represents the combination of two different genes, \(R\) and \(W\), from each parent plant. Using the Binomial Theorem, the polynomial expansion can be calculated by evaluating each term formed by the expression. This not only simplifies the expression but also helps visualize the different possible gene pairings in the offspring.

The expansion of \((R + W)^2\) uses specific coefficients from the binomial series, making it easier to understand how different powers of \(R\) and \(W\) contribute to the final result.

Genetics

In genetics, the process of inheritance and expression of traits can often be modeled using polynomial expressions. Here, \(R\) and \(W\) are used to denote distinct gene variants, or alleles. The expression \((R + W)^2\) visually represents the probability and combinations of these alleles being passed down from parents to offspring.

When you expand this expression, you see all possible ways these genes can pair.

• \(R^2\) represents both parent plants passing the \(R\) gene.
• \(2RW\) represents one \(R\) and one \(W\) gene being combined.
• \(W^2\) represents both parent plants passing the \(W\) gene.

The expanded polynomial \(R^2 + 2RW + W^2\) shows the potential genetic makeup of the offspring. Each term corresponds to how genes can pair up to produce specific traits.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the terms of the expanded form of a binomial raised to a power, thanks to the Binomial Theorem. They are represented as \( \binom{n}{k} \), and they help determine how the terms are weighed in the expansion.

For the expression \((R + W)^2\), we use the coefficients \(\binom{2}{0} = 1\), \(\binom{2}{1} = 2\), and \(\binom{2}{2} = 1\). These values are derived from Pascal's Triangle and are used to multiply the respective terms in the expansion.

Understanding these coefficients is essential because they provide insight into the probability or frequency of each gene pairing occurring. In the context of genetics, it tells us that the combination \(2RW\) is twice as likely to occur as \(R^2\) or \(W^2\). These coefficients help us predict the likelihood of different offspring gene combinations.