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Chapter 5: Problem 43

Find each product. \((x+1)^{3}\)

Short Answer

Expert verified
The expanded form is \(x^3 + 3x^2 + 3x + 1\).

Step by step solution

01

Recognize the Binomial Theorem

To find the product of \(x+1\)^3, use the binomial theorem which states \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).Here, \(a = x\), \(b = 1\), and \(n = 3\).
02

Expand Using the Binomial Coefficients

Apply the binomial theorem: \((x+1)^3 = \binom{3}{0} x^3 + \binom{3}{1} x^2 \cdot 1 + \binom{3}{2} x \cdot 1^2 + \binom{3}{3} \cdot 1^3\).
03

Simplify Each Term

Calculate each binomial coefficient: \ \binom{3}{0} = 1, \ \ \binom{3}{1} = 3, \ \ \binom{3}{2} = 3, \ and \ \binom{3}{3} = 1 \.Thus, the expanded form is \(x^3 + 3x^2 + 3x + 1\).
04

Combine the Terms

Write the final result: \(x^3 + 3x^2 + 3x + 1\).

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Key Concepts

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

polynomial expansion
Polynomial expansion is a way of writing polynomials as a sum of terms with exponents. For example, expanding \( (x + 1)^3 \) involves expressing it as \( x^3 + 3x^2 + 3x + 1 \). The expansion process uses the binomial theorem, which helps us break down expressions like \( (a + b)^n \) into a series of simpler terms. Each term in the expanded form of a polynomial has a coefficient and variables raised to different powers. The goal of polynomial expansion is to make complex expressions easier to work with.
binomial coefficients
Binomial coefficients are the numerical factors that multiply the terms in polynomials expanded using the binomial theorem. They are denoted by \(\binom{n}{k}\), read as 'n choose k'. These coefficients can be found in Pascal's Triangle or calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). In the exercise given, we see four binomial coefficients: \( \binom{3}{0} = 1, \binom{3}{1} = 3, \binom{3}{2} = 3, \binom{3}{3} = 1 \). They determine how many ways we can choose k items from n items without regard to order. When expanding \( (x+1)^3 \), these coefficients help to distribute the powers of \( x \) and \( 1 \) correctly across the expansion.
algebraic expressions
Algebraic expressions are combinations of letters and numbers joined by operations like addition, subtraction, multiplication, and division. They include variables (like \( x \)) and constants (like 1). These expressions can be simplified or expanded using various algebraic rules and theorems. In the provided exercise, \( (x+1)^3 \) is an algebraic expression that was expanded using the binomial theorem. The result, \( x^3 + 3x^2 + 3x + 1 \), is also an algebraic expression but in its expanded form. Simplifying or expanding algebraic expressions helps make them easier to understand and manipulate when solving equations or working with polynomials.

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Most popular questions from this chapter

Assume that \(a\) is a number greater than 1 . Arrange the following terms in order from least to greatest: \(-(-a)^{3},-a^{3},(-a)^{4},-a^{4} .\) Explain how you decided on the order.

In Objective \(I,\) we showed how \(6^{\circ}\) acts as 1 when it is applied to the product rule, thus motivating the definition of 0 as an exponent. We can also use the quotient rule to motivate this definition. Consider the expression \(\frac{25}{25} .\) What is its simplest form?

Use scientific notation to calculate the answer to each problem. In \(2007,9.63 \times 10^{9}\) dollars were spent to attend motion pictures in the United States. Domestic admissions (the total number of tickets sold) for that year totaled 1.4 billion. What was the average ticket price?

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}} $$

Write each product as a sum of terms. Write answers with positive exponents only. Simplify each term. See Section 1.8 $$ \frac{1}{4 y}\left(y^{4}+6 y^{2}+8\right) $$
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