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Chapter 6: Problem 7

Divide and check. $$ \frac{36 x^{6}+18 x^{5}-27 x^{2}}{9 x^{2}} $$

Short Answer

Expert verified
The simplified expression is \[ 4 x^4 + 2 x^3 - 3 \].

Step by step solution

01

- Separate the terms

Write the given expression as separate fractions for each term in the numerator over the denominator: \[ \frac{36 x^6}{9 x^2} + \frac{18 x^5}{9 x^2} - \frac{27 x^2}{9 x^2} \]
02

- Simplify each fraction

For each term, divide the coefficients and apply the quotient rule for exponents: \[ \frac{36 x^6}{9 x^2} = 4 x^{6-2} = 4 x^4 \] \[ \frac{18 x^5}{9 x^2} = 2 x^{5-2} = 2 x^3 \] \[ \frac{27 x^2}{9 x^2} = 3 x^{2-2} = 3 \]
03

- Combine simplified terms

Combine the simplified terms to obtain the final expression: \[ 4 x^4 + 2 x^3 - 3 \]
04

- Check the solution

To check the solution, multiply the simplified expression by the original denominator to see if it equals the original numerator: \[ (4 x^4 + 2 x^3 - 3) \times 9 x^2 = 36 x^6 + 18 x^5 - 27 x^2 \] The left side of the equation matches the numerator, confirming the solution is correct.

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Key Concepts

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

quotient rule for exponents
When dividing polynomials, one of the most crucial rules is the quotient rule for exponents. This rule states that when you divide two expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
For example, \[\frac{36 x^6}{9 x^2}\] can be simplified using this rule. Here are the steps:
  • Divide the coefficients: \[\frac{36}{9} = 4\]
  • Subtract the exponents: \[x^{6-2} = x^4\]
So, \[\frac{36 x^6}{9 x^2} = 4 x^4\]. This simplification makes complex polynomial division clear and manageable.
Applying this rule effectively allows for easy handling of even the most troublesome polynomial division problems.
simplifying fractions
Simplifying fractions is critical in polynomial division. It helps make the terms easier to manage and combine. We start by breaking a complex fraction into simpler parts, separating each term in the numerator by the denominator. For instance, the expression \[\frac{36 x^6 + 18 x^5 - 27 x^2}{9 x^2}\] becomes:
  • \[\frac{36 x^6}{9 x^2}\]
  • \[\frac{18 x^5}{9 x^2}\]
  • \[\frac{27 x^2}{9 x^2}\]
Next, each fraction is simplified individually:
  • \[\frac{36 x^6}{9 x^2} = 4 x^4\]
  • \[\frac{18 x^5}{9 x^2} = 2 x^3\]
  • \[\frac{27 x^2}{9 x^2} = 3\]
By separating and simplifying, we transform a daunting expression into manageable parts. This simplification is essential for understanding and solving polynomial division problems.
checking division results
After simplifying a polynomial division problem, checking your work is essential. This process verifies that the solution is correct. To check, multiply the simplified expression by the original denominator and see if it equals the original numerator.
For the given problem, we simplified the expression to \[(4 x^4 + 2 x^3 - 3)\]. To check this, multiply it by the original denominator \[(9 x^2)\]:
\[(4 x^4 + 2 x^3 - 3) \times 9 x^2\]
Distributing \[(9 x^2)\]:
  • \[4 x^4 \times 9 x^2 = 36 x^6\]
  • \[2 x^3 \times 9 x^2 = 18 x^5\]
  • \[-3 \times 9 x^2 = -27 x^2\]
Combining these terms, we get the original numerator: \[(36 x^6 + 18 x^5 - 27 x^2)\]. Since both sides match, the solution is correct. Always check your division to ensure accuracy.

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