Problem 30 - 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. $$(t+13)^{2}$$

Step-by-Step Solution

Answer

The expanded form is \(t^2 + 26t + 169\).

Step 1: Identify the Binomial

The expression given is \[(t+13)^2\]which is a binomial raised to the power of 2.

Step 2: Recall the Binomial Square Formula

For a binomial \((a + b)^2\), the square formula is:\[(a + b)^2 = a^2 + 2ab + b^2\]This formula allows us to expand the binomial.

Step 3: Apply the Binomial Square Formula

In the given expression, \(a = t\) and \(b = 13\). Substitute these values into the formula:\[(t + 13)^2 = t^2 + 2(t)(13) + 13^2\]

Step 4: Calculate Each Term

1. \(t^2 = t^2\),2. \(2(t)(13) = 26t\),3. \(13^2 = 169\).

Step 5: Combine the Terms

Write the expanded expression by combining the terms:\[t^2 + 26t + 169\]

Key Concepts

Binomial Square Formula

The binomial square formula is a shortcut used to square a binomial, which is a type of algebraic expression consisting of two terms. When you see an expression like \((a + b)^2\), it simply means you are squaring the binomial \((a + b)\). The binomial square formula is given by:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This formula helps simplify calculations, avoiding the need to multiply the whole expression twice manually. For instance, given \((t + 13)^2\), it can be quickly expanded using the formula:

• \(t^2 + 2(t)(13) + 13^2\)
• Which gives \(t^2 + 26t + 169\)

Using the binomial square formula makes calculations quicker and reduces errors, especially in longer expressions.

Special Patterns in Algebra

In algebra, recognizing special patterns can significantly simplify the process of solving equations and expressions. The binomial square formula is one of these special patterns, allowing for quick expansion of squared binomials. Similarly, what makes these patterns special is their consistent and predictable result.

• A common pattern is the difference of squares: \(a^2 - b^2 = (a+b)(a-b)\)
• Another is the perfect square trinomials, like \((a+b)^2\) and \((a-b)^2\)

Mastering these patterns means you can recognize them in various forms and situations, making the work faster and less prone to mistakes. Spotting these patterns amidst complex algebraic expressions helps in decomposing and solving them efficiently, transforming lengthy multiplication steps into a more direct process.

Binomials in Algebra

Binomials are a foundational element in algebra. They consist of two terms, often connected by a plus \((+)\) or minus \((-\)) sign, and can be part of more extensive algebraic expressions. Examples include expressions like \(t+13\) or \(x-5\).Understanding how to manipulate binomials is crucial because they frequently appear in equations, inequalities, and various algebraic contexts. This manipulation might include operations like addition, subtraction, or, as we've seen, squaring a binomial using the binomial square formula.Recognizing these pairs and understanding operations on them is vital for multiple areas in mathematics, such as:

• Solving quadratic equations, as binomials are often part of these equations
• Factoring complex polynomials, which sometimes require breaking down into recognizable binomial expressions
• Graphing polynomials, where binomials form linear or quadratic terms to be graphed

Overall, mastering binomials equips students with essential skills to tackle more advanced algebraic problems, providing a solid foundation for further mathematical studies.