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Chapter 4: Problem 64

Find each product. $$ (t+2)^{2} $$

Short Answer

Expert verified
The expanded form is \( t^{2} + 4t + 4 \).

Step by step solution

01

Understand the Problem

The exercise requires finding the product of \( (t+2)^{2} \). This is a binomial expression raised to the second power.
02

Use the Binomial Theorem

Recall that \( (a+b)^{2} \) can be expanded to \( a^{2} + 2ab + b^{2} \). Here, \( a = t \) and \( b = 2 \).
03

Substitute and Expand

Substitute \( t \) and \( 2 \) into the formula: \( (t+2)^{2} = t^{2} + 2 \cdot\ t \cdot\ 2 + 2^{2} \).
04

Simplify the Expression

Simplify the terms: \( t^{2} + 4t + 4 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. These symbols represent numbers and quantities in formulas and equations.

In algebra, expressions like \(t + 2\) can represent a relationship. When you encounter problems like \( (t + 2)^{2}\), you enter more advanced algebra. Here, knowing how to expand and simplify these expressions is vital.

Understanding the distributive property is essential. Take \(a(b + c) = ab + ac\). The distribution works similarly when squaring a binomial. Practice these principles to master algebra and solve expressions like \( (t + 2)^{2}\).
Quadratic Expressions
Quadratic expressions are polynomial expressions of degree 2. They typically take the form \(ax^2 + bx + c\). When expanding \( (t + 2)^{2}\), you deal with a quadratic expression.

In the problem \( (t + 2)^{2} = t^2 + 4t + 4\), the expression results from squaring a binomial. The term \(t^2\) is the quadratic term, \(4t\) is the linear term, and \(4\) is the constant term.

To understand it better, let's break it down:
  • The quadratic term \(t^2\) comes from multiplying \(t \cdot t\).
  • The linear term \(4t\) is from \(2 \cdot t \cdot 2\).
  • The constant term \(4\) originates from \(2^2\).
By working through each term, you can grasp how to form and simplify quadratic expressions.
Binomial Theorem
The Binomial Theorem deals with the expansion of expressions raised to a power. A binomial is an algebraic expression with two terms such as \(a + b\). When raised to a power (like \( (a + b)^n\)), it expands in a specific pattern.

The formula for squaring a binomial \( (a + b)^2 = a^2 + 2ab + b^2\) simplifies expressions like \( (t + 2)^2\). Here's how it works for the given problem:
  • \

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Most popular questions from this chapter

Find each product. $$ [(5 x+1)+6 y]^{2} $$

A student subtracted \(\left(15 x^{2}+12\right)-\left(7 x^{2}-3\right)\) vertically as follows. $$\begin{array}{r}15 x^{2}+12 \\\7 x^{2}-3 \\\\\hline 8 x^{2}+9\end{array}$$ Give the correct difference.

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(3 r^{2}\right)^{2} r^{-5}}{r^{-2} r^{3}}\left(2 r^{-6}\right)^{2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(k^{2}\right)^{-3} k^{4} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{m^{2} n^{-8}}{m^{-6} n^{3}} $$
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