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Chapter 0: Problem 51

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

Short Answer

Expert verified
The simplified form of \( (x+1)^{3} \) is \( x^{3} + 3x^{2} + 3x + 1 \).

Step by step solution

01

Consider the Expression

Consider expression \( (x + 1)^{3} \). This is a binomial, where \( x \) and \( 1 \) are the two terms, and the power is 3.
02

Expand the Expression

Expand \( (x + 1)^{3} \) as \( (x + 1) * (x + 1) * (x + 1) = x^{3} + 3x^{2} + 3x + 1 \). This step involves the application of the binomial theorem which gives \( (a+b)^{n}= a^{n} + na^{n-1}b+ n(n-1)a^{n-2}b^{2}/{2!} \ldots + b^{n} \).
03

Simplify the Result

On combining like terms, we obtain the final simplified result, which is \( x^{3} + 3x^{2} + 3x + 1 \).

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Key Concepts

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

Polynomial Expansion
When dealing with expressions involving polynomials, one often encounters the need to expand them. Polynomial expansion involves writing a polynomial in its most extended form by multiplying out all of its terms.
This process can help to make subsequent operations, such as integration or derivation, simpler.
In the case of expanding a polynomial like - \[(x+1)^3\], this means expressing it as the sum of its terms without parentheses.The degree of each term in the polynomial plays a crucial role in determining how the polynomial behaves and interacts with other functions.
Binomial Theorem
The binomial theorem is a powerful tool that allows you to expand expressions of the form - \((a+b)^n\) into a sum involving terms of the form - \(a^kb^{n-k}\).Using the binomial theorem- the expansion formula can be applied, which in this example takes the expression \((x + 1)^3\) and systematically breaks it down into: - \(x^3 + 3x^2 \cdot 1 + 3x \cdot 1^2 + 1^3\).This theorem provides a systematic and quick way of expanding binomials without the need for repeated multiplication, especially as the power, \( n \), becomes large.
Like Terms
Whenever you expand a polynomial or binomial, you often end up with "like terms." These are terms that have the same variable raised to the same power. For example, in - \(x^3 + 3x^2 + 3x + 1\),- each term has a degree, but none are like terms since all have distinct powers of \(x\).
Combine like terms to simplify expressions.
This results in reducing expressions to their simplest form.
Simplification
Simplification is the process of reducing complex expressions into simpler or more manageable forms. This involves consolidating like terms, combining constants, and ensuring there are no unnecessary brackets or exponents. In the polynomial expansion of \((x + 1)^3\), after applying the binomial theorem, you check if any terms can be combined to make the expression easier to handle.
In our example: - \(x^3 + 3x^2 + 3x + 1\) - there is no further simplification needed because none of the terms can be grouped further.

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