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Chapter 5: Problem 32

Add the polynomials. $$\left(7 r^{4}+5 r^{2}+2 r\right)+\left(-18 r^{4}-5 r^{2}-r\right)$$

Short Answer

Expert verified
The result of adding the given polynomials is \(-11r^4 + r\).

Step by step solution

01

Identify Similar Terms

The similar terms in the polynomials are those that have the same variable with the same exponent. Here, they are \(7 r^{4}\) and \(-18 r^{4}\) for \(r^4\), \(5 r^{2}\) and \(-5 r^{2}\) for \(r^2\), and \(2 r\) and \(-r\) for \(r\).
02

Adding the Terms with Same Exponent of r

Add the coefficients of the terms we identified earlier to simplify. This gives us \(7r^4 + (-18r^4) = -11r^4\), \(5r^2 + (-5r^2) = 0r^2\), and \(2r + (-r) = r\). Since \(0r^2\) equals to zero, it can be omitted.
03

Combine the Results

Finally, add all the resulting terms together to get the simplified version of the input polynomial. So, the final polynomial is \(-11r^4 + r\).

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Key Concepts

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

Like Terms
When working with polynomial expressions, understanding the concept of "Like Terms" is crucial. Like Terms are terms that share the same variables raised to the same power. Identifying them is the first step in simplifying polynomials. In the given exercise, you are provided with terms such as \(7r^4\) and \(-18r^4\). These are like terms because they both possess the variable \(r\) with the exponent 4. Similarly, \(5r^2\) and \(-5r^2\) are like terms because they each contain \(r^2\). Recognizing these terms allows us to group and combine them effectively.
  • Always check for the same variable and exponent.
  • Like Terms simplify polynomial calculations.
  • Grouping Like Terms is beneficial in obtaining a cleaner expression.
With like terms identified, you can proceed towards simplifying the polynomial by combining these terms properly.
Polynomial Simplification
Polynomial simplification involves reducing a polynomial to its simplest form. This means that you combine like terms and ensure there are no unnecessary terms left. In our problem, once like terms are identified, the process of simplification becomes straightforward.
For example, combine \(7r^4\) and \(-18r^4\) by adding their coefficients:
  • This results in \(-11r^4\).
  • The \(5r^2\) and \(-5r^2\) cancel each other out, resulting in a zero term \(0r^2\), which we can omit.
  • Combine \(2r\) and \(-r\) to end up with \(r\).
Understanding and applying polynomial simplification streamlines the expression to its most concise form, making it easier to work with in mathematical calculations.
Combining Terms
Combining terms in polynomials refers to the process of bringing together all like terms to form a simplified expression. After identifying like terms, the next step is to perform arithmetic operations on their coefficients. In our exercise, by adding the coefficients for the same powers of \(r\), we effectively combine terms.
Here's how you combine them:
  • First, add the coefficients of like terms: for example, \(7 + (-18) = -11\).
  • Apply this to each power of the variable: \(5 - 5 = 0\) for \(r^2\), \(2 - 1 = 1\) for \(r\).
  • Write out the simplified polynomial resulting from these operations.
By combining terms, we simplify the polynomial to \(-11r^4 + r\), eliminating all redundant terms and ensuring a simple, understandable expression. This makes polynomial operations manageable and less prone to errors, enhancing clarity in your math work.

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Most popular questions from this chapter

Perform the indicated computations. Express answers in scientific notation. $$\left(2 \times 10^{2}\right)\left(2.6 \times 10^{-3}\right) \div\left(4 \times 10^{3}\right)$$

How do you know if a number is written in scientific notation?

In Exercises \(100-103,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a polynomial in \(x\) of degree 6 is divided by a monomial in \(x\) of degree \(2,\) the degree of the quotient is 4

In Exercises \(85-86,\) the variable \(n\) in each exponent represents a natural Number. Divide the polynomial by the monomial. Then use polynomial multiplication to check the quotient. $$\frac{12 x^{15 n}-24 x^{12 n}+8 x^{3 n}}{4 x^{3 n}}$$

In Exercises \(53-78,\) divide the polynomial by the monomial. Check each answer by showing that the product of the divisor and the quotient is the dividend. $$\frac{20 x^{7} y^{4}-15 x^{3} y^{2}-10 x^{2} y}{-5 x^{2} y}$$
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