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Chapter 4: Problem 53

Add $$\begin{array}{r}3 m^{2}+5 m+6 \\\\+\left(2 m^{2}-2 m-4\right) \\\\\hline\end{array}$$

Short Answer

Expert verified
5m^2 + 3m + 2

Step by step solution

01

Identify like terms

First, recognize the terms that have the same variable and power. Here the like terms are those containing: \(m^2\), \(m\), and constants.
02

Add the coefficients of the \(m^2\) terms

Combine the coefficients of the \(m^2\) terms: \(3m^2\) and \(2m^2\). Thus, \(3 + 2 = 5\). So, the resulting term is \(5m^2\).
03

Add the coefficients of the \(m\) terms

Combine the coefficients of the \(m\) terms: \(5m\) and \(-2m\). Thus, \(5 - 2 = 3\). So, the resulting term is \(3m\).
04

Add the constant terms

Combine the constant terms: \(6\) and \(-4\). Thus, \(6 - 4 = 2\). So, the resulting term is \(2\).
05

Write the final simplified polynomial

Combine all the resulting terms together to get the final answer: \(5m^2 + 3m + 2\).

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Key Concepts

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

Like Terms
When adding polynomials, the first thing you need to do is identify the like terms. Like terms are terms that have the same variable raised to the same power. For example, in the exercise given, the terms with the variable and power combination of \(m^2\) are \(3m^2\) and \(2m^2\). Similarly, the terms \(5m\) and \(-2m\) have the same variable \(m\) raised to the first power. Constants are also considered like terms. Identifying like terms is crucial as it sets the stage for combining them correctly.
Coefficients
A coefficient is the numerical part of a term that is multiplied by the variable. For example, in the term \(3m^2\), the number 3 is the coefficient. In polynomial addition, you add or subtract the coefficients of the like terms. In the exercise, the \(m^2\) terms \(3m^2\) and \(2m^2\) have coefficients 3 and 2, respectively. By adding these coefficients, we get \(3 + 2 = 5\), resulting in the term \(5m^2\). Understanding coefficients is essential for correctly performing operations on polynomials.
Combining Terms
Combining terms is the process of adding or subtracting the coefficients of like terms. After identifying like terms and their coefficients, you simply add or subtract the numbers. In the exercise, we combined the \(m\) terms \(5m\) and \(-2m\). By subtracting the coefficients (since \(-2\) is a negative number), we get \(5 - 2 = 3m\). Similarly, we combined the constants \(6\) and \(-4\) to get \(2\). Finally, we combine all resulting terms to get the simplified polynomial \(5m^2 + 3m + 2\). This step puts everything together for the final answer.

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Most popular questions from this chapter

Find each product. $$ (3 t-4 s)(t+3 s) $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications. Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 101 \times 99 $$

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If an object is projected upward under certain conditions, its height in feet is given by the trinomial $$-16 t^{2}+60 t+80$$ where \(t\) is in seconds. Evaluate this trinomial for \(t=2.5 .\) Use the result to fill in the blanks: If ________ seconds have elapsed, then the height of the object is ____________ feet.

\(\frac{\left(9^{-1} z^{-2} x\right)^{-1}\left(4 z^{2} x^{4}\right)^{-2}}{\left(5 z^{-2} x^{-3}\right)^{2}}\)
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