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Chapter 5: Problem 22

Add the polynomials. $$\left(-3 x^{2}+x\right)+\left(4 x^{2}+8 x\right)$$

Short Answer

Expert verified
\(x^2 + 9x\)

Step by step solution

01

Identify Like Terms

In the given polynomials, the like terms are terms that have the same variables and powers. The like terms in the two polynomials are \(-3x^2\) and \(4x^2\), and \(x\) and \(8x\).
02

Addition of Like Terms

Next, add the coefficients of like terms. Adding the coefficients of \(-3x^2\) and \(4x^2\) results in \(x^2\), and adding \(x\) and \(8x\) gives \(9x\).
03

Write the Result

Combine the results of the addition of like terms to form the resulting polynomial. So, the resulting polynomial is \(x^2 + 9x\).

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Key Concepts

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

Like Terms
In the world of polynomials, "like terms" play a crucial role in simplifying expressions. Like terms are terms that have the same variable raised to the same power. This means they share identical components except for their coefficients. Identifying like terms is a pivotal step in combining polynomials. For example, in the polynomials
  • off. This is because they each include the variable raised to the power of 2. Similarly, the terms
    • are also like terms since they share the same variable and power even if the variables are not accompanied by exponents. Recognizing these terms allows us to streamline our calculations by focusing our efforts on similar components of each polynomial expression. By organizing like terms together, polynomial addition becomes much more manageable.
Coefficient Addition
The next step is to perform coefficient addition, which involves adding the numerical coefficients of each pair of like terms. This step is integral as it simplifies the polynomial, making it easier to interpret and solve.
For the like terms,
  • addition of the coefficients When you add these coefficients, you essentially combine the like terms into a single term for each variable and power combination. This process requires attention to detail as it's easy to make minor errors if the coefficients are miscalculated. Always ensure that you are only adding the coefficients of terms that are truly like terms.
Resulting Polynomial
After conducting coefficient addition, the resulting polynomial is formed through the combination of newly calculated terms. This resulting polynomial retains all the variables and exponents but with simplified coefficients. The result is a much cleaner and consolidated polynomial.
  • In our earlier example, we began with two polynomials, This signifies the culmination of a careful and systematic process where every term is checked and then combined. The advantage of forming a resulting polynomial is that it is simpler, making it easier to work with in further mathematical operations or application contexts. As with any mathematics problem, it's always a good practice to double-check each step to ensure the correctness of the resulting polynomial.

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Most popular questions from this chapter

Express \(\frac{7}{8}\) as a decimal. (Section \(1.3,\) Example 4 )

List the whole numbers in this set: $$\left\\{-4,-\frac{1}{5}, 0, \pi, \sqrt{16}, \sqrt{17}\right\\}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$(y-1)\left(y^{2}+y+1\right)=y^{3}-1$$

Use a vertical format to find each product. $$\begin{array}{r}7 x^{3}-5 x^{2}+6 x \\\3 x^{2}-4 x \\\\\hline\end{array}$$

Exercises \(110-112\) will help you prepare for the material covered in the next section. In each exercise, perform the long division without using a calculator, and then state the quotient and the remainder. $$1 9 \longdiv { 4 9 4 }$$
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