Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, expand the binomial \((9 b+1)^{2}\)

Step-by-Step Solution

Answer

The expansion of \((9b + 1)^2\) is \(81b^2 + 18b + 1\).

Step 1: Apply the Binomial Theorem for Squaring

To expand the expression \((9b + 1)^2\), use the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). In this case, \(a = 9b\) and \(b = 1\).

Step 2: Calculate the Square of the First Term

Calculate \(a^2 = (9b)^2 = 81b^2\).

Step 3: Calculate the Double Product of the Terms

Calculate \(2ab = 2 \times 9b \times 1 = 18b\).

Step 4: Calculate the Square of the Second Term

Calculate \(b^2 = 1^2 = 1\).

Step 5: Combine All Terms to Form the Expanded Expression

Substitute the calculated values back into the expanded form: \(81b^2 + 18b + 1\). This is the expansion of \((9b + 1)^2\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool that allows us to expand expressions that are raised to any power. It is especially useful for expanding binomials, which are expressions consisting of two terms. For example, expressions like

• \((a + b)^n\)

can be expanded into a sum of terms involving coefficients, powers of the first term, and powers of the second term. The coefficients are known as binomial coefficients and can be found using Pascal's triangle or by calculating combinations.
When you square a binomial, you're essentially applying the principles of the Binomial Theorem, though in a more simplified form because the power is just 2.

Squaring a Binomial

Squaring a binomial is a specific application of the binomial expansion, where the exponent is 2. The formula you need to remember when squaring a binomial

• \((a + b)^2\)

is

• \(a^2 + 2ab + b^2\)

This formula simplifies the process, making it quick to calculate:

• First, square the first term: \(a^2\).
• Next, double the product of the two terms: \(2ab\).
• Finally, square the last term: \(b^2\).

For instance, for the binomial \((9b + 1)^2\), the squared terms are calculated step by step, resulting in \(81b^2 + 18b + 1\). Each operation respects the order dictated by the formula, ensuring accuracy.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. Variables often represent unknown values and can be manipulated using algebraic rules. Understanding algebraic expressions is crucial because they form the basis for crafting equations and performing operations like addition, subtraction, and multiplication of polynomials.
For example, in the expansion of \((9b + 1)^2\), each part—like \(9b\)—is considered an algebraic term. Algebraic expressions become polynomial expressions when they have more than one term. Mastering the manipulation of these expressions allows for the solution of more complex problems and equations.

Polynomial Expansion

Polynomial expansion is the process of breaking down expressions raised to powers into simpler components or terms. This is often necessary for simplifying expressions or further solving problems involving them. With polynomial expansions, the complexity of an expression can be reduced to a sum of individual terms, each described by a product of coefficients and variables raised to specific powers.
Consider the operation on \((9b + 1)^2\): by expanding this using the formula for squaring a binomial, you transform a compact expression into a polynomial \(81b^2 + 18b + 1\). This expanded form is free from parentheses and offers a clearer perspective on each contributing part, readily interpreted or further manipulated in algebraic equations. Understanding these expansions is fundamental in algebra and lays the groundwork for more advanced math topics.