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Chapter 5: Problem 3

For Problems \(1-24\), divide the monomials. $$ \frac{4 x^{3}}{2 x} $$

Short Answer

Expert verified
\( 2x^2 \)

Step by step solution

01

Identify the Terms

We have the division of two monomials: \( \frac{4x^3}{2x} \). Here, 4 and 2 are the coefficients, and \( x^3 \) and \( x \) are the variables with their respective exponents.
02

Divide the Coefficients

Divide the coefficients (numbers) first: \( \frac{4}{2} = 2 \).
03

Apply the Quotient Rule for Exponents

According to the quotient rule for exponents: \( \frac{x^a}{x^b} = x^{a-b} \). For the given problem, apply this to the variables: \( \frac{x^3}{x} = x^{3-1} = x^2 \).
04

Combine the Results

Combine the results from the previous steps. The coefficient result is 2, and the exponent result is \( x^2 \). Therefore, the simplified expression is \( 2x^2 \).

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Key Concepts

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

Monomial Division
Monomial division is the process of dividing one monomial by another. A monomial is a single term algebraic expression like \(4x^3\) or \(2x\). Each monomial consists of a coefficient (a constant number) and a variable raised to a power (exponent). In our example \(\frac{4x^3}{2x}\), 4 and 2 are coefficients, and \(x^3\) and \(x\) indicate the variable with its exponents.
  • To divide monomials, you separately divide their coefficients.
  • Then, handle the variable part using the exponents, which involves another useful rule.
This clear two-part process helps simplify expressions and makes complex problems more manageable.
Quotient Rule for Exponents
The quotient rule for exponents is a key tool for dividing variables in monomials. It states that when you divide two powers with the same base, you subtract the exponents: \(\frac{x^a}{x^b} = x^{a-b}\). Applying it can greatly simplify the variable part of a division.
Consider our problem \(\frac{x^3}{x}\) as an example:
  • Both terms have the base \(x\).
  • Subtract the exponent of the denominator from the exponent of the numerator: \(x^{3-1} = x^2\).
This rule allows for easy and systematic processing of expressions, especially when variables and their powers are involved.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means reducing them to their simplest form, which often results in fewer terms or a more concise expression. For expressions like the one given, it involves several calculations: focusing on each part with care, coefficients and variables.
Steps typically involve:
  • Dividing coefficients, which are the numbers in front of variables. Here, 4 divided by 2 gives us 2.
  • Using the quotient rule for exponents (as previously discussed) to manage the variable parts.
  • Combining the simplified components into one neat expression. In our example, this gives us \(2x^2\).
By following these methods, any complex expression can frequently be simplified, aiding in further mathematical operations or problem-solving.

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Most popular questions from this chapter

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (4 a+3)(3 a-4) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (x-14)(x+8) $$

For Problems \(11-36\), find the indicated products by applying the distributive property and combining similar terms. Use the following format to show your work: $$ \begin{aligned} (x+3)(x+8) &=x(x)+x(8)+3(x)+3(8) \\ &=x^{2}+8 x+3 x+24 \\ &=x^{2}+11 x+24 \end{aligned} $$ $$ \left(x^{2}-3 x+4\right)\left(x^{2}+5 x-2\right) $$

For Problems \(113-120\), find the indicated products. Don't forget that \((x+2)^{3}\) means \((x+2)(x+2)(x+2)\). $$ (4 n-3)^{3} $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (n-5)(n-9) $$
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