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Chapter 13: Problem 39

Find each sum. $$\left(4 t-t^{2}\right)+(8 t+2)$$

Short Answer

Expert verified
The sum is \(-t^2 + 12t + 2\).

Step by step solution

01

Write Down the Expression

The given expression is \((4t - t^2) + (8t + 2)\). It is composed of two separate expressions that need to be combined.
02

Remove the Parentheses

Since addition is associative, you can remove the parentheses to rewrite the expression as \(4t - t^2 + 8t + 2\).
03

Combine Like Terms

Identify and combine like terms: Coefficients of \(t^2\), \(t\), and the constants separately. First, notice there is only one \(t^2\) term, so it stays as \(-t^2\). Next, combine \(4t\) and \(8t\) to get \(12t\). Finally, the constant term is \(2\). This gives \(-t^2 + 12t + 2\).

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Key Concepts

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

Understanding Like Terms
In algebra, a key concept is identifying and working with 'like terms'. Like terms have exactly the same variables raised to the same power. For example, in the expression \(4t - t^2 + 8t + 2\), the terms \(4t\) and \(8t\) are like terms.Why? Because both are terms involving the variable \(t\) to the first power. Meanwhile, \(-t^2\) has the same variable \(t\), but it is raised to the second power, so it is not a like term with \(4t\) and \(8t\).Constants, like the number \(2\) in this expression, don't have variables attached and stand alone. They are considered like terms with other constants. Remember:
  • Like terms can be combined through addition or subtraction.
  • Unalike terms must be kept separate and cannot be combined.
Demystifying Algebraic Expressions
An algebraic expression is essentially a mathematical phrase that can include numbers, variables (letters that stand in for unknown or changing quantities), and operations (such as addition and multiplication). In the example from the exercise, \((4t - t^2) + (8t + 2)\), you encounter two algebraic expressions that are being added together.Expressions do not have an equality sign, unlike equations. Solving an expression involves simplifying it, as opposed to finding a specific numerical answer unless values for variables are given.To simplify an expression:
  • First, rearrange it by removing parentheses, if necessary, using properties of operations like associativity and commutativity.
  • Next, identify and combine like terms to achieve the simplest form.
This helps in accurately solving problems and understanding relationships between terms.
Combining Polynomials in Simple Steps
Combining polynomials is a technique used to simplify expressions that involve multiple polynomials. In the given exercise, you combine two polynomials: \(4t - t^2\) and \(8t + 2\).To combine, follow these simple steps:
  • Remove any grouping symbols like parentheses, as they are not necessary when adding or subtracting polynomials, due to the associative property of addition.
  • Identify like terms by their variables and powers.
  • Add or subtract the coefficients of like terms.
In our problem, remove the parentheses to form the expression \(4t - t^2 + 8t + 2\). Recognize that \(4t\) and \(8t\) are like terms, and add their coefficients: \(4 + 8 = 12\). The expression becomes \(-t^2 + 12t + 2\).Simplifying expressions by combining polynomials is a foundational skill that makes solving complex algebraic problems more manageable.

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Most popular questions from this chapter

Determine whether each equation or table represents a linear or nonlinear function. Explain. $$\begin{array}{|r|r|} \hline x & y \\ \hline-4 & 12 \\ \hline-2 & 0 \\ \hline 0 & 4 \\ \hline 2 & 0 \\ \hline \end{array}$$

Seth is making a rectangular dog pen. He needs the length of the pen to be 5 feet less than 4 times the width. Write an expression for the area \(A\) of the dog pen. A \(10 x-10 \mathrm{ft}^{2}\) B \(4 x^{2}-5 x \mathrm{ft}^{2}\) C \(4 x-5 \mathrm{ft}^{2}\) D \(x^{2}-5 x \mathrm{ft}^{2}\)

Find each product. $$n(n+6)$$

Find each difference. $$(4 x+y)-(5 x+y)$$

Use a table to graph each line. $$y=-x$$
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