/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Use the binomial formula to expa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 30

Use the binomial formula to expand each binomial. $$(3+2 a)^{4}$$

Short Answer

Expert verified
The expanded form is \(81 + 216a + 216a^2 + 96a^3 + 16a^4\).

Step by step solution

01

Identify the Binomial Formula

The binomial formula for expansion is given by the expression: \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\] where \(n\) is the exponent, \(\binom{n}{k}\) is the binomial coefficient, \(x\) is the first term in the binomial, and \(y\) is the second term.
02

Set the Values

For the expression \((3+2a)^4\), set \(x = 3\), \(y = 2a\), and \(n = 4\). We will use these values to expand the binomial using the formula.
03

Calculate the Binomial Coefficients

Calculate the binomial coefficients \(\binom{4}{k}\) for \(k = 0\) to \(4\):- \(\binom{4}{0} = 1\)- \(\binom{4}{1} = 4\)- \(\binom{4}{2} = 6\)- \(\binom{4}{3} = 4\)- \(\binom{4}{4} = 1\)
04

Apply the Binomial Formula

Substitute the coefficients and terms into the binomial formula: - For \(k=0\): \(\binom{4}{0} (3)^{4} (2a)^{0} = 1 \cdot 81 \cdot 1 = 81\)- For \(k=1\): \(\binom{4}{1} (3)^{3} (2a)^{1} = 4 \cdot 27 \cdot 2a = 216a\)- For \(k=2\): \(\binom{4}{2} (3)^{2} (2a)^{2} = 6 \cdot 9 \cdot 4a^2 = 216a^2\)- For \(k=3\): \(\binom{4}{3} (3)^{1} (2a)^{3} = 4 \cdot 3 \cdot 8a^3 = 96a^3\)- For \(k=4\): \(\binom{4}{4} (3)^{0} (2a)^{4} = 1 \cdot 1 \cdot 16a^4 = 16a^4\)
05

Write the Expanded Form

Combine all the terms together to write the expanded form of the binomial: \[(3 + 2a)^4 = 81 + 216a + 216a^2 + 96a^3 + 16a^4\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Binomial Theorem
The binomial theorem is a crucial mathematical tool that allows us to expand expressions that are raised to a power, specifically binomials. A binomial is an expression containing two terms. The formula for the binomial theorem is expressed as:
  • The general form: \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
  • The summation symbol \(\sum\) indicates that you add up all the terms from \(k = 0\) to \(k = n\).
  • The binomial coefficient \(\binom{n}{k}\) is calculated using combinations, which can be found using factorial notation.
Understanding this theorem helps in solving problems efficiently, especially in simplifying expressions in algebra and calculus. It's notable for the pattern it reveals in powers of binomials.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables (like \(a\), \(b\)) and operators (such as +, -, *, /). Algebraic expressions are the building blocks of algebra, helping us model real-world situations and solve various equations.
When dealing with binomials, the expression often includes:
  • Constants: Numbers with fixed values, such as the '3' in our given binomial \((3+2a)^4\).
  • Variables: Symbols that represent numbers or values, like 'a' in the expression.
  • Coefficients: Numbers that multiply the variables, indicated by '2' in '2a'.
Understanding each part of an algebraic expression is the key to appropriately expanding or simplifying it using rules like those found in the binomial theorem.
Polynomial Expansion
Polynomial expansion involves expressing a binomial raised to a power as a sum of terms, each consisting of a coefficient, variable parts, and constants. The expanded polynomial is often easier to use in further calculations and provides insight into the nature of the expression.
In the example \((3+2a)^4\), polynomial expansion through the binomial theorem leads us to:
  • Multiple terms: It results in five terms: \(81, 216a, 216a^2, 96a^3, \text{and }16a^4\).
  • Simplified computation: By expanding, we can easily compute specific values or solve equations involving the polynomial.
  • Insight into terms: The expansion shows the contribution of each individual part of the binomial to the final polynomial.
Consequently, polynomial expansion is vital in both theoretical and applied mathematics, making complex algebraic manipulations more approachable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Multiply. See Section 5.4 $$ (x-2)^{2} $$

Solve. See Example 5 In free fall, a parachutist falls 16 feet during the first second, 48 feet during the second second, 80 feet during the third second, and so on. Find how far she falls during the eighth second. Find the total distance she falls during the first 8 seconds.

Solve. Find the sum of the first twenty terms of the sequence \(\frac{1}{2}, \frac{1}{4}, 0, \ldots,-\frac{17}{4}\) where \(-\frac{17}{4}\) is the twentieth term.

Evaluate. See Section \(1.3 .\) $$ 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 $$

A person has a choice between two job offers. Job \(A\) has an annual starting salary of \(\$ 30,000\) with guaranteed annual raises of \(\$ 1200\) for the next four years, whereas job \(B\) has an annual starting salary of \(\$ 28,000\) with guaranteed annual raises of \(\$ 2500\) for the next four years. Compare the fifth partial sums for each sequence to determine which job would pay more money over the next 5 years.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.