Problem 29 - 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. $$(t+9)^{2}$$

Step-by-Step Solution

Answer

The expanded form of \\( (t+9)^2 \\\) is \\( t^2 + 18t + 81 \\\).

Step 1: Identify the Expression Type

The given expression \( (t+9)^2 \) is a binomial squared. Recall that the square of a binomial follows the pattern \( (a+b)^2 = a^2 + 2ab + b^2 \). Here, identify \( a = t \) and \( b = 9 \).

Step 2: Apply the Binomial Square Pattern

Using the pattern derived from \( (a+b)^2 \), we substitute \( a \) and \( b \) into the formula: \((t+9)^2 = t^2 + 2(t)(9) + 9^2\).

Step 3: Calculate Each Component

Perform the individual calculations: - Calculate \( t^2 \) which remains \( t^2 \).- Calculate \( 2(t)(9) = 18t \).- Calculate \( 9^2 = 81 \).

Step 4: Combine the Results

Combine all the calculated components to arrive at the final expanded expression: \( t^2 + 18t + 81 \).

Key Concepts

Understanding Binomial Squared

When you see something like \((t+9)^2\), you're dealing with a binomial squared. A binomial is simply an algebraic expression that contains two terms. In this case, the two terms are \(t\) and \(9\). Squaring a binomial involves multiplying the binomial by itself. However, you can simplify the process by using a special pattern.

• The special pattern for the square of a binomial \((a+b)^2\) is \(a^2 + 2ab + b^2\).
• This means you square the first term, double the product of both terms, and then square the second term.

Understanding this pattern makes calculations faster and helps prevent mistakes. Instead of expanding it by traditional multiplication, this shortcut gives you the answer more directly.

Breaking Down Algebraic Expressions

An algebraic expression is made up of numbers, variables, and operators. In our problem, \((t+9)^2\) is the original algebraic expression we're working with. To handle an expression like this, it's important to understand what each part represents.

• \(t\) is a variable, which can represent any number.
• \(9\) is a constant term, meaning it does not change.
• The parenthesis and the exponent show that the entire expression inside the parenthesis needs to be squared.

By dissecting the expression, we can see how parts interact under different operations like addition, multiplication, and exponentiation. In our task, we used these components and worked with them individually according to the rules of the binomial square pattern.

Special Patterns in Algebra

Algebra is full of special patterns that simplify complex calculations. Recognizing these patterns allows us to work smarter, not harder. For example, the binomial square pattern \((a+b)^2 = a^2 + 2ab + b^2\) is one such shortcut that provides a quick way to expand squared binomials without manual multiplication.

• These patterns are like shortcuts that help solve problems faster and with less risk of error.
• Similar patterns exist for the difference of squares and trinomial expansions.

By learning these patterns, students can handle a wide range of algebraic expressions with efficiency. In this exercise, the binomial square pattern was used to solve for the product of \((t+9)^2\), resulting in \(t^2 + 18t + 81\). Knowing these patterns not only aids in solving specific problems but also builds a strong foundation for more advanced algebra.