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Chapter 4: Problem 51

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

Short Answer

Expert verified
The result is \(5x^2 - 2x\).

Step by step solution

01

Write down the expressions

The two expressions to be added are: \[2x^2 - 4x\] and \[3x^2 + 2x\].
02

Combine like terms

Identify the like terms in each expression. Like terms are those with the same variable raised to the same power. Here, combine the coefficients of like terms:- For the term with \(x^2\): \(2x^2 + 3x^2\)- For the term with \(x\): \(-4x + 2x\).
03

Add the coefficients

Perform the addition for each set of like terms:- For \(x^2\) terms: \(2x^2 + 3x^2 = 5x^2\)- For \(x\) terms: \(-4x + 2x = -2x\).
04

Write the final expression

Combine the results from Step 3 to get the final expression: \[5x^2 - 2x\].

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Key Concepts

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

like terms
In algebra, 'like terms' are terms that have the same variables raised to the same power. For example, in the expressions \(2x^2 - 4x\) and \(3x^2 + 2x\), the like terms are \(2x^2\) and \(3x^2\) because they both involve \(x^2\). Similarly, \(-4x\) and \(2x\) are like terms because they both involve \(x\). Identifying like terms is crucial because it allows us to combine them into a single term. This process simplifies our algebraic expressions and makes calculations easier. In any given polynomial, always look for like terms to combine them effectively.
combining coefficients
Combining coefficients means adding or subtracting the numerical parts (coefficients) of like terms. When we identify like terms, we can simplify the expression by combining their coefficients. For example, considering the terms \(2x^2\) and \(3x^2\), we combine the coefficients (2 and 3) to get \(5x^2\). Similarly, for \(-4x\) and \(2x\), we combine the coefficients (-4 and 2) to get \(-2x\). This step is essential when adding or subtracting polynomials. By combining coefficients, we reduce the complexity of the expression, making it easier to work with.
algebraic expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations (like addition, subtraction, multiplication, and division). An example of an algebraic expression is \(2x^2 - 4x + 3\). These expressions can be simplified by combining like terms. In the context of polynomial addition, we start with individual algebraic expressions and combine them to form a simpler overall expression. For instance, given the problem \(2x^2 - 4x + (3x^2 + 2x)\), we simplify it by combining like terms to get \(5x^2 - 2x\). Understanding how to handle algebraic expressions is foundational for solving more complex algebra problems.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$ \begin{aligned} &y=(x+3)^{2}\\\ &\begin{array}{c|c|c|c|c|c} x & -5 & -4 & -3 & -2 & -1 \\ \hline y & & & & & \end{array} \end{aligned} $$

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. $$ 201 \times 199 $$

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}-1\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m+6)^{2} $$

Add or subtract as indicated. \((4 x+2 x y-3)-(-2 x+3 x y+4)\)
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