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Chapter 6: Problem 38

Expand each binomial. $$ (3 x+2 y)^{4} $$

Short Answer

Expert verified
The expansion of \( (3 x+2 y)^{4} \) is \( 81x^4 + 432x^3*y + 216x^2*y^2 + 48x*y^3 + 16y^4 \).

Step by step solution

01

Recognize the form

Recognize that \( (3 x+2 y)^{4} \) is the power of a binomial. Therefore, we can use the Binomial Theorem to expand it.
02

Apply the Binomial Theorem

Applying the Binomial Theorem leads to the following sequence of computations: \( (3x+2y)^4 = 4C0*(3x)^4*(2y)^0 + 4C1*(3x)^3*(2y)^1 + 4C2*(3x)^2*(2y)^2+ 4C3*(3x)^1*(2y)^3 + 4C4*(3x)^0*(2y)^4 \). Here, 'C' stands for the binomial coefficient, and '4C0', '4C1', etc. are specific binomial coefficients.
03

Calculate and simplify

Now, just calculate each term and simplify: \( 81x^4 + 4*108x^3*y + 6*36x^2*y^2 + 4*12x*y^3 + 16y^4 \). This simplifies to \( 81x^4 + 432x^3*y + 216x^2*y^2 + 48x*y^3 + 16y^4 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem
The Binomial Theorem is a key mathematical principle for expanding expressions that are raised to a power. It provides a systematic way to expand binomials, which are expressions with two terms, such as \[(a + b)^n\]Using the Binomial Theorem, you can express the expansion in terms of combinations:
  • Each term in the expansion is a combination of the form \( nCk \), which represents the number of ways to select \( k \) items from \( n \) total items.
  • The formula for the coefficients is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
  • Each term takes the form: \[\binom{n}{k}a^{n-k}b^{k}\]

For example, when expanding \[(3x + 2y)^4\],we use the Binomial Theorem coefficients and powers of \(3x\) and \(2y\) to find each term.
It's essential because it helps simplify complex algebraic processes and solves various problems in combinatorics and probability.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition and multiplication. They are foundational in algebra and beyond, used to represent real-world scenarios and unknown quantities. In the context of our problem, the expression \((3x + 2y)^4\) is an algebraic expression consisting of a binomial raised to a power. Here:
  • \(3x\) and \(2y\) are terms that include coefficients and variables.
  • Raising the expression to a power involves multiplying it by itself a specified number of times鈥攊n our case, four times.

Simplifying such expressions requires following specific rules, such as distribution and combining like terms, which are essential skills in algebra. Understanding how to manipulate and work with algebraic expressions is crucial for solving equations and inequalities effectively.
Polynomial Expansion
Polynomial expansion is the process of expressing a polynomial that is raised to a power in its expanded form. A polynomial is an expression with multiple terms added together, often with varying powers of variables.During polynomial expansion, each term of a binomial is multiplied individually, and the results are summed up to give the full expression. For example, expanding \((3x + 2y)^4\) results in a polynomial with terms of different degrees.
  • The degree of a term is the sum of the exponents of the variables within it. For instance, in the term \(81x^4\), the degree is 4.
  • The terms \(432x^3y\), \(216x^2y^2\), \(48xy^3\), and \(16y^4\) arise from multiplying instances of \(3x\) and \(2y\) with coefficients provided by the Binomial Theorem.

This expansion allows for easier computation and understanding of the behavior of polynomials. It plays a significant role in calculus, numerical methods, and other advanced mathematical applications.

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Most popular questions from this chapter

Expand each binomial. $$ (x+2)^{5} $$

Find the specified term of each binomial expansion. Third term of \((x-2)^{12}\)

Write each function in vertex form. $$ y=-4 x^{2}+9 $$

Use the Binomial Theorem to expand each binomial. $$ (w+1)^{5} $$

A box has 10 items, and you select 3 of them. What is the value of \(P-C\) if \(P\) represents the number of permutations possible when selecting 3 of the items, and \(C\) is the number of combinations possible when selecting 3 of the items?
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