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Chapter 5: Problem 16

Divide and check. $$\frac{9 x^{2}+3 x-2}{3 x}$$

Short Answer

Expert verified
The result is \[ 3x + 1 - \frac{2}{3x} \]

Step by step solution

01

- Set Up the Division

Write the given expression \( \frac{9x^2 + 3x - 2}{3x} \) as a fraction to be divided.
02

- Divide Each Term by the Denominator

Divide each term in the numerator \( 9x^2, 3x, \text{and} -2 \) by the denominator \( 3x \). \[ \frac{9x^2}{3x} + \frac{3x}{3x} - \frac{-2}{3x} \]
03

- Simplify Each Term

Simplify each term: \[ \frac{9x^2}{3x} = 3x \, \, \frac{3x}{3x} = 1 \, \, \frac{-2}{3x} = - \frac{2}{3x} \]
04

- Combine the Terms

Combine the simplified terms together from step 3 to get the final answer: \[ 3x + 1 - \frac{2}{3x} \]
05

- Verification

Verify by multiplying the denominator back with the result to check if we get the original numerator: \[ (3x + 1 - \frac{2}{3x}) \times 3x = 9x^2 + 3x - 2 \]

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Key Concepts

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

Polynomial Division
Polynomial division is a method used to divide one polynomial by another. It's like long division with numbers, but instead, we work with variables and coefficients. In our exercise, we are dividing the polynomial \(9x^2 + 3x - 2\) by the monomial \(3x\). The main goal is to simplify the expression.
To start:
  • Write the given expression as a fraction: \( \frac{9x^2 + 3x - 2}{3x} \).
  • Divide each term in the numerator by the denominator:
  • Split the fraction into separate parts: \( \frac{9x^2}{3x} + \frac{3x}{3x} - \frac{-2}{3x} \).

This setup helps us simplify the expression step by step.
Simplification
Simplifying the expression means reducing it to its simplest form. Simplification makes it easier to understand and work with polynomials.

Next, we simplify each term:
  • \( \frac{9x^2}{3x} = 3x \): Divide the coefficients (9 divided by 3 equals 3) and subtract the exponents (2 - 1 = 1).
  • \( \frac{3x}{3x} = 1 \): Anything divided by itself is 1.
  • \( \frac{-2}{3x} \): This stays as it is, a fraction.

After simplifying each term, we combine them:
\( 3x + 1 - \frac{2}{3x} \).
This step-by-step simplification makes the problem more manageable.
Verification Steps
Verifying our result ensures that we didn't make any mistakes during the division and simplification process.

To verify:
  • Multiply the simplified result \(3x + 1 - \frac{2}{3x}\) by the denominator \(3x\).


Let's do this step-by-step:
  • First, distribute \(3x\) to each term: \(3x(3x) + 3x(1) - 3x\left(\frac{2}{3x}\right)\).
  • Next, perform the multiplication: \(9x^2 + 3x - 2\).

You can see we get back the original polynomial in the numerator. This confirms our division and simplification were correct.
Verification is crucial because it confirms the accuracy of our solution.
It helps catch any errors and ensures our final answer is reliable.

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