/*! 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 31 Expand each binomial. $$ (x+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 31

Expand each binomial. $$ (x+1)^{6} $$

Short Answer

Expert verified
\((x + 1)^6 = x^{6} + 6x^{5} + 15x^{4} + 20x^{3} + 15x^{2} + 6x + 1\).

Step by step solution

01

Applying the Binomial Theorem

We can start by applying the binomial theorem for our problem. The equation we use from the binomial theorem is also known as the binomial expansion: \((x + 1)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} 1^{k}\). Considering 1 raised to any power is 1, we can simplify this equation to: \((x + 1)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k}\).
02

Performing the Expansion

Next, we substitute values of \(k\) from 0 to 6 into the equation. The coefficients \(\binom{6}{k}\) are computed by 6 Choose k (also known as Combination). The simplified expansion is: \((x + 1)^6 = \binom{6}{0}x^{6} + \binom{6}{1}x^{5} + \binom{6}{2}x^{4} + \binom{6}{3}x^{3} + \binom{6}{4}x^{2} + \binom{6}{5}x^{1} + \binom{6}{6}x^{0}\).
03

Simplification of the Expansion

In this step, we simply have to compute the binomial coefficients and rewrite the equation. \((x + 1)^6 = x^{6} + 6x^{5} + 15x^{4} + 20x^{3} + 15x^{2} + 6x + 1\).

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 powerful tool in algebra that provides a method to expand expressions raised to any given power. When you see expressions like \((x + 1)^n\), the Binomial Theorem allows you to express this expanded form efficiently. Essentially, it uses the formula: \[(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^{k}\] where \(n\) is any positive integer, and \(\binom{n}{k}\) are binomial coefficients.

In our original exercise, exploring \((x + 1)^6\) involves applying this theorem with \(a = 1\) and \(n = 6\). Each term in this expansion corresponds to a value of \(k\) from 0 to \(n\), showing a blended mix of powers of \(x\) and the constants.
Binomial Coefficients
Binomial coefficients are integral to working through binomial expansions. They arise from the expression \(\binom{n}{k}\), which is read as "n choose k" and represents combinations in mathematics. This counts the number of ways to choose \(k\) elements from \(n\) elements without considering the order.

For the expansion of \((x + 1)^6\), these coefficients, \(\binom{6}{0}, \binom{6}{1}, \binom{6}{2}, \ldots, \binom{6}{6}\), are easily computed by specific formulas or Pascal鈥檚 Triangle. Each coefficient corresponds to a unique term in the expression, influencing its associated power of \(x\). The computed coefficients are 1, 6, 15, 20, 15, 6, and 1, respectively.
Algebraic Expressions
Algebraic expressions refer to combinations of variables, numbers, and operators in math. In the expansion of a binomial using the Binomial Theorem, we break down complex expressions into simpler parts.

For example, when you expand \((x + 1)^6\), it becomes a lengthy expression comprising different powers of \(x\), each with specific coefficients. This expanded form is made of terms like \(x^6\), \(6x^5\), and so on, each contributing to a complete representation of the original problem.

Understanding algebraic expressions and their proper handling allows for a straightforward simplification process, aiding you in solving wider mathematical problems.

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

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

Expand each binomial. $$ (x+5 y)^{3} $$

In the sequence \(1 !, 2 !, 3 !, 4 !, 5 !, 6 !, \ldots,\) the first term that ends with a zero is \(5 ! .\) a. Explain why \(5 !\) and all the terms following \(5 !\) end with a zero. b. Find the number of zeros with which \(100 !\) ends.

Probability A coin is tossed ten times. The probability of heads on each toss is \(0.5 .\) Evaluate each probability. a. exactly 5 heads \(\quad\) b. exactly 6 heads \(\quad\) c. exactly 7 heads

Write each polynomial in standard form. Then classify it by degree and by number of terms. $$ -3 x^{2}+6-x^{3} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.