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Chapter 8: Problem 71

Perform the indicated operations. \((5+9 x)+(-4-8 x)\)

Short Answer

Expert verified
1 + x

Step by step solution

01

Identify Like Terms

Separate the constants and the variable terms from both expressions. The expression contains constants 5 and -4, and variable terms 9x and -8x.
02

Combine Like Terms

Add the constants and the variable terms separately. Combine the constants: 5 + (-4) = 1. Combine the variable terms: 9x + (-8x) = x.
03

Write the Final Expression

Combine the results from the previous step to form the simplified expression: 1 + x.

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Key Concepts

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

Combining Like Terms
Combining like terms is a fundamental concept when working with polynomials. Like terms are terms that have the same variable raised to the same power. For example, in the expression (5 + 9x) + (-4 - 8x), the like terms are:
  • The constants: 5 and -4
  • The variable terms: 9x and -8x
To combine these like terms:
1. Add the constants together: 5 + (-4) = 1
2. Add the variable terms together: 9x + (-8x) = x
It鈥檚 essential to remember that you can only combine terms that have identical variables and exponents. So, 2x and 3x虏 are not like terms and cannot be directly combined.
Polynomial Simplification
Polynomial simplification often involves combining like terms, as seen previously. This step-by-step process reduces an expression to its simplest form, making it easier to work with. To illustrate:
1. Start with the given polynomial expression: (5 + 9x) + (-4 - 8x)
2. Identify and group like terms, as we did: constants (5 and -4) and variable terms (9x and -8x).
3. Combine the grouped terms:
  • Constants: 5 + (-4) = 1
  • Variable terms: 9x + (-8x) = x
4. Assemble these simplified terms to form the final expression: 1 + x
Simplifying polynomials makes further operations, such as addition, subtraction, or even polynomial division, more straightforward to perform.
Variable Terms
Variable terms are parts of the polynomial that include a variable (like x) combined with a coefficient (a number in front of the variable). For example, in 9x, 9 is the coefficient and x is the variable. Understanding variable terms involves recognizing a few key points:
  • Variable terms can be similar or different. Similar variable terms must have the same variable and exponent, like 2x and 3x.
  • When combining variable terms, you only combine their coefficients and keep the variable part the same.
In our example (5 + 9x) + (-4 - 8x), the variable terms are 9x and -8x. To simplify, we add the coefficients of these like terms: 9 + (-8) = 1, resulting in x. Effective handling of variable terms is crucial for polynomial operations and further algebraic problem-solving.

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Most popular questions from this chapter

Combine like terms. $$ 7 m^{5}-2 m^{3}+8 m^{5}-m^{3} $$

Find each quotient. $$ \frac{-7+4 i}{3+2 i} $$

Multiply. $$ (2-i)^{2}(2+i)^{2} $$

Write with rational exponents, and then apply the properties of exponents. Assume that all radicands represent posititive real mumbers. Give answers in exponential form. See Example 6. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) since, when \(a=1\) and \(b=0\) a true statement results: $$ \sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b $$ Explain why this is faulty reasoning.

Find the distance between each pair of points. $$ (-1,2) \text { and }(5,3) $$
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