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Chapter 4: Problem 6

Divide. $$ \frac{27 m^{4}-18 m^{3}-9}{9} $$

Short Answer

Expert verified
\(3 m^{4} - 2 m^{3} - 1\)

Step by step solution

01

Set Up the Division

Identify the polynomial in the numerator \[27 m^{4} - 18 m^{3} - 9\] and the constant in the denominator \[9\].
02

Divide Each Term

Divide each term of the polynomial by the constant. \[\frac{27 m^{4}}{9} - \frac{18 m^{3}}{9} - \frac{9}{9}\]
03

Simplify Each Term

Simplify the division for each term: \[\frac{27 m^{4}}{9} = 3 m^{4}\]\[\frac{18 m^{3}}{9} = 2 m^{3}\]\[\frac{9}{9} = 1\]
04

Combine the Simplified Terms

Write the simplified terms together to get the final expression: \[3 m^{4} - 2 m^{3} - 1\]

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Key Concepts

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

Division of Polynomials
Dividing polynomials involves breaking down a polynomial expression by another polynomial or a constant. In this exercise, we are dividing by a constant. This process involves:

First, identifying the polynomial in the numerator, in this case, \[27 m^{4} - 18 m^{3} - 9\], and the constant in the denominator, \[9\].
Then, you need to divide every term in the polynomial by that constant.
This step-by-step approach helps manage possibly complex expressions by turning the polynomial division into simpler manageable calculations.
Lastly, combine all terms to get the final simplified polynomial.
Simplifying Polynomials
Simplifying polynomials is about reducing them to their most basic form. Here鈥檚 a closer look at how to simplify each term after you have set up the division:

  • Take each term of the polynomial divided by the constant separately.
  • Perform the simplification for every term in the expression. For instance:
    \[\frac{27 m^{4}}{9} = 3 m^{4}\]\beginladderand,

    \[\frac{18 m^{3}}{9} = 2 m^{3}\]and,
    \[\frac{9}{9} = 1\].

Combining these simplified terms, you get the final simplified form of the polynomial: \[3 m^{4} - 2 m^{3} - 1\].
By systematically simplifying each part, polynomial division becomes a simpler task.
Algebraic Expressions
An algebraic expression represents a mathematical phrase that can include numbers, variables, and operators.
In the division of polynomials, these play a crucial role.

  • In our specific problem, the algebraic expression is \[27 m^{4} - 18 m^{3} - 9\] divided by \[9\].
  • Understanding the algebraic structure makes it clearer how each term interacts when divided by the constant.

Thanks to well-defined rules in algebra, such divisions and simplifications follow a predictable pattern.
This example is a great illustration of simplifying complex-looking expressions into standard polynomial forms.

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Most popular questions from this chapter

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Find each product. $$ (2 a+b)\left(3 a^{2}+2 a b+b^{2}\right) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{(3 p q) q^{2}}{6 p^{2} q^{4}} $$

Find each product. $$ (2 p+7)^{2} $$
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