The expression $(2a - 3b)^4$ represents the fourth power of the binomial $(2a - 3b)$, meaning it is the result of multiplying $(2a - 3b)$ by itself four times. This term is a prime candidate for application of the binomial theorem, which provides a formula to expand binomials raised to a positive integer power. The expansion yields a sum of terms that involves the coefficients and powers of each variable in the binomial.
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Using the binomial theorem, $(x + y)^n$ can be expanded as \(\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\), allowing us to find each term's coefficient in the expansion.
In this case, applying the theorem will yield terms that contain powers of both $2a$ and $-3b$, resulting in various combinations.
The coefficient for each term in the expansion can be calculated using binomial coefficients based on their respective position in the expansion.
The final expanded form will include both positive and negative terms due to the subtraction in the binomial $(2a - 3b)$.
Each term in the expansion is formed by taking different combinations of $(2a)$ and $(-3b)$ raised to their respective powers, ensuring that the total power sums up to 4.
Review Questions
How does the binomial theorem facilitate the expansion of $(2a - 3b)^4$, and what role do coefficients play in this process?
The binomial theorem allows us to expand $(2a - 3b)^4$ by providing a systematic way to calculate each term's coefficient based on its position in the expansion. Each term in the expansion involves binomial coefficients, which determine how many times a particular combination of powers appears. For example, for any given term with $x$ from $(2a)$ and $y$ from $(-3b)$, the coefficient can be calculated using \(\binom{4}{k}\) where $k$ is the number of times $(-3b)$ is chosen. This results in a precise representation of all possible combinations derived from multiplying $(2a - 3b)$ four times.
In what ways does Pascal's Triangle provide insight into finding coefficients when expanding $(2a - 3b)^4$?
Pascal's Triangle visually represents binomial coefficients that are essential for expanding expressions like $(2a - 3b)^4$. Each row corresponds to an increasing power of a binomial, with coefficients indicating how many ways terms can combine. For instance, row four corresponds to the expansion of a binomial raised to the fourth power and provides direct access to coefficients such as 1, 4, 6, 4, and 1 for use in expanding our expression. Understanding this triangle allows for quick reference and calculation without needing factorials.
Evaluate how understanding the properties of polynomials enhances your ability to work with expressions like $(2a - 3b)^4$ and predict their characteristics.
Understanding polynomial properties helps predict outcomes when dealing with expressions like $(2a - 3b)^4$. Polynomials have specific behaviors regarding degree, leading coefficients, and roots. By recognizing that this expression is a polynomial of degree four with leading term $16a^4$, one can infer that there will be multiple terms resulting from combining various powers of $2a$ and $-3b$. This knowledge aids in predicting characteristics like end behavior and possible zeroes, enhancing overall comprehension and problem-solving skills when working with complex polynomial expressions.
Related terms
Binomial Coefficient: A numerical coefficient that appears in the binomial expansion, calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), representing the number of ways to choose k elements from a set of n elements.
A triangular array of numbers where each number is the sum of the two directly above it, which can be used to find binomial coefficients for expansions.
Polynomial: An expression consisting of variables raised to whole number powers and coefficients, including terms like $(2a - 3b)^4$ which is a specific type of polynomial.