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Chapter 5: Problem 47

Combine like terms. Write all answers in descending order. \(4 b^{3}+5 b+7 b^{3}+b^{2}-6 b\)

Short Answer

Expert verified
The simplified polynomial is \(11b^{3} + b^{2} - b\).

Step by step solution

01

Identify like terms

Look at the given polynomial: \(4b^{3}+5b+7b^{3}+b^{2}-6b\). Identify terms with the same variable and exponent. In this case, the like terms are: \(4b^{3}\) and \(7b^{3}\), \(5b\) and \(-6b\), and \(b^{2}\) which does not have any like terms.
02

Combine like terms for \(b^{3}\)

Add the coefficients of the like terms \(4b^{3}\) and \(7b^{3}\).\(4b^{3} + 7b^{3} = 11b^{3}\)
03

Combine like terms for \(b\)

Next, add the coefficients of the like terms \(5b\) and \(-6b\).\(5b - 6b = -1b = -b\)
04

Combine all terms

Now, combine all the simplified terms: \(11b^{3}\), \(b^{2}\), and \(-b\). Write them in descending order of exponents: \(11b^{3} + b^{2} - b\)

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Key Concepts

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

Polynomials
Polynomials are algebraic expressions that consist of terms added or subtracted together. Each term is made up of a coefficient, which is a number, and a variable raised to a power, like in the term \(4b^3\). A polynomial can have one or multiple terms, such as in our example: \(4b^3 + 5b + 7b^3 + b^2 - 6b\). In a polynomial, the highest exponent of the variable is known as the degree. Understanding the structure of polynomials helps in combining like terms and simplifying the expression.
Like Terms
Like terms are terms in a polynomial that have the same variable raised to the same power. For instance, in the polynomial \(4b^3 + 5b + 7b^3 + b^2 - 6b\), the terms \(4b^3\) and \(7b^3\) are like terms because they both have the variable \(b\) raised to the third power. Likewise, \(5b\) and \(-6b\) are like terms since they both contain the variable \(b\) to the power of one. However, \(b^2\) does not have any like terms in this example. Combining like terms means adding or subtracting their coefficients.
Descending Order
Descending order refers to arranging the terms of a polynomial so that the exponents decrease from left to right. For our example, the final simplified polynomial \(11b^3 + b^2 - b\) is written in descending order. Here, the term \(11b^3\) comes first because it has the highest exponent, followed by \(b^2\) and finally \(-b\), which has the smallest exponent. Writing in descending order helps to standardize the format and make comprehension and further manipulation of the polynomial easier.

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Most popular questions from this chapter

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{-4 p^{8}}{3 m^{2} n^{3}}\right)^{3} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -3+(-11) $$

Find the domain of \(F / G,\) if $$ F(x)=\frac{1}{x-4} \quad \text { and } \quad G(x)=\frac{x^{2}-4}{x-3} $$

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For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{1}{2-x}\\\ &g(x)=7-x \end{aligned} $$
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