Problem 66 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(x+1)^{3}$$

Step-by-Step Solution

Answer

The product \\(x+1\\)^{3} expands to \\(x^3 + 3x^2 + 3x + 1\\).

Step 1: Recognize the Expression Pattern

The given expression \(x+1\)^{3} is a cube of a binomial. We will use the binomial theorem or the special cube expansion formula to simplify it.

Step 2: Recall the Binomial Cube Expansion Formula

The cube of a binomial \(a + b\)^3 is expanded using the formula: \(a^3 + 3a^2b + 3ab^2 + b^3\). Here, \a = x\ and \b = 1\.

Step 3: Substitute and Simplify Terms

Substitute \a = x\ and \b = 1\ into the formula: \(x^3 + 3x^2 \cdot 1 + 3x \cdot 1^2 + 1^3\). Calculate each term separately:1. \x^3\ is \x^3\.2. \3x^2 \cdot 1 = 3x^2\.3. \3x \cdot 1^2 = 3x\.4. \1^3 = 1\.

Step 4: Write Final Expanded Form

Combine all the terms:\(x^3 + 3x^2 + 3x + 1\). This is the expanded form of \(x+1\)^{3}.

Key Concepts

Binomial Expansion

Binomial expansion is a mathematical method used to expand expressions that are raised to a power. When you see something like \((a + b)^n\),you're dealing with a binomial expression to the power of\(n\).The binomial theorem provides a powerful formula to help expand these expressions without multiplying them out the long way.

• Essentially, binomial expansion allows us to break down binomials elevated to any specified power.
• This is especially useful for simplifying or calculating polynomial expressions.

For any binomial\((a + b)^n\),the expression can be expanded as:\[\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\]This formula involves coefficients known as binomial coefficients, which are calculated using the formula:\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).Each term in the expanded binomial corresponds to a different value of\(k.\)Understanding these coefficients is crucial because they determine the multiplier for each term in the expanded form.

Cube of a Binomial

Cubing a binomial refers to raising a binomial expression to the third power. The expression \((a + b)^3\)can be expanded in a special way without needing to multiply manually three times.

• The binomial cube expansion formula is given as:\(a^3 + 3a^2b + 3ab^2 + b^3.\)
• This formula is derived from either using the binomial theorem or repeated multiplication.
• The expanded form of cubing a binomial includes four terms, each systematically calculated.

Using this formula simplifies what would otherwise be a lengthy and detailed multiplication process.
For instance, when cubing \((x + 1)\),you substitute\(a = x\) and\(b = 1\)to get:\[x^3 + 3x^2 \cdot 1 + 3x \cdot 1^2 + 1^3.\]After simplifying each term, the expanded form for this binomial is:\(x^3 + 3x^2 + 3x + 1.\)

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation like addition, subtraction, multiplication, or division. They form the basis for many algebra problems and solutions.

• Expressions such as \((x + 1)\)are considered algebraic because they use variables (like\(x\)).
• When algebraic expressions are combined through operations, they become more complex, such as \((x + 1)^3\).

Algebraic expressions can be simplified or expanded using various mathematical techniques. Binomial expansion is one of these techniques, which specifically helps when dealing with expressions raised to a power involving two terms.
Understanding how to work with algebraic expressions is crucial because it allows us to solve and simplify many types of equations. Mastery of these expressions strengthens mathematical problem-solving skills and aids in comprehending more complex mathematical concepts.