Problem 96 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(t-11)^{2}$$

Step-by-Step Solution

Answer

The short answer for the given expression \((t-11)^2\) is \(t^2 - 22t + 121\).

Step 1: Identify the terms in the binomial#

In the given expression \((t-11)^2\), we have \(a = t\) and \(b = 11\).

Step 2: Apply the square of a binomial formula#

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), we substitute the values of \(a\) and \(b\). In this case, \(a = t\) and \(b = 11\).

Step 3: Substitute and expand the expression#

Applying the formula, we get: \((t - 11)^2 = t^2 - 2(t)(11) + 11^2\)

Step 4: Simplify the expression#

Now, we just need to multiply and simplify the terms: \(t^2 - 22t + 121\) So, the expanded form of the given expression is \((t-11)^2 = t^2 - 22t + 121\).

Key Concepts

Binomial Expansion

In algebra, a binomial is simply an expression that contains two terms, like \((a + b)\) or \((t - 11)\). When we expand a binomial, we multiply out the terms to express it as a polynomial. This process is known as binomial expansion. It often involves using formulas to break apart and simplify expressions. The most common use of binomial expansion in problems is in operations involving powers, like squares or cubes. For example, expanding \((t - 11)^2\) requires using a formula that helps distribute the squared power correctly over each part of the binomial. This formula is:

• \((a - b)^2 = a^2 - 2ab + b^2\)

This formula represents how each component of the binomial contributes to the final polynomial. Understanding it allows us to tackle even more complex expressions down the line. It's a building block in algebra that helps widen our comprehension of polynomial expressions and their simplifications.

Squaring Binomials

Squaring a binomial is a specific type of binomial expansion that occurs when you raise a binomial to the power of 2. The goal when squaring a binomial is to expand and then simplify the expression. Let's consider the binomial \((t-11)\). When squaring this binomial, as shown in the original exercise, we apply the square of a binomial formula:

• For \((a - b)^2\), the result is \(a^2 - 2ab + b^2\).

First, take each part of the binomial and apply the formula terms:

• \(a^2\) becomes \(t^2\).
• \(-2ab\) becomes \(-2 \times t \times 11\), resulting in \(-22t\).
• \(b^2\) becomes \(11^2\), which simplifies to \(121\).

Putting these results together transitions the expression from \((t-11)^2\) to the expanded form \(t^2 - 22t + 121\). This method reduces errors and ensures a straightforward pathway from binomial to polynomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. These expressions can be as simple as \(3x\) or as complex as \((t-11)^2\). They are fundamental in algebra as they form the basis for equations, which we use to solve mathematical problems involving unknowns. Each part of an algebraic expression has a name:

• The number in front of a variable is the coefficient.
• The variable represents an unknown value.
• Constants are numbers on their own, without variables.

In \(t^2 - 22t + 121\):

• \(t^2\) is a term where \(t\) is a variable and 1 is its coefficient.
• \(-22t\) has the variable \(t\) with a coefficient of -22.
• \(121\) is a constant term.

Understanding these components is crucial for manipulating and simplifying expressions. This knowledge will assist in solving equations and resolving complex algebraic problems with confidence. Algebra flows into nearly every branch of mathematics, making comprehension of these fundamental concepts vital for advanced study.