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Chapter 5: Problem 5

Divide and check. $$\frac{32 x^{5}-24 x}{8}$$

Short Answer

Expert verified
The simplified form is 4\(x^{5}\) - 3\(x\).

Step by step solution

01

Simplify the Numerator

First, factor out the greatest common factor (GCF) from the numerator. The terms in the numerator are 32\(x^5\) and -24\(x\). The GCF is 8. Therefore, \[\frac{32 x^{5}-24 x}{8} = \frac{8(4 x^{5} - 3 x)}{8}\].
02

Cancel out the Common Factor

Next, simplify by dividing the numerator and the denominator by the common factor. Since both the numerator and the denominator contain 8, cancel them out:\[\frac{8(4 x^{5} - 3 x)}{8} = 4 x^{5} - 3 x\].
03

Verify your Solution

Check if the simplified form is correct by distributing the denominator back into the simplified answer. Multiply 8 by each term in the simplified form to see if you get back to the original numerator:\[8(4 x^{5} - 3 x) = 32 x^{5} - 24 x\].This confirms that our simplification is correct.

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Key Concepts

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

Greatest Common Factor
Understanding the Greatest Common Factor (GCF) is crucial in simplifying polynomial expressions. The GCF is the largest number or term that is a factor of each term in the given expression. For instance, in the expression 32\(x^5\) - 24\(x\), both terms share a common factor of 8. Identifying this helps streamline the process of polynomial division.
To identify the GCF in more complex expressions:
  • Factorize each term individually.
  • Pick the highest shared factor among all terms.
This makes it easier to simplify expressions by factoring out the GCF.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form, making them easier to work with. In polynomial division, this is done by identifying and canceling out common factors. For example, given the expression \[\frac{32x^{5}-24x}{8}\], we start by identifying the GCF, which is 8. Then we factor this out from the numerator, resulting in \[\frac{8(4x^{5} - 3x)}{8}\].
Next, simplifying involves canceling the common factor in both the numerator and the denominator. Here, the 8s cancel each other out, leaving us with 4\(x^5\) - 3\(x\). By repeatedly practicing this, one can quickly identify patterns and reduce expressions efficiently.
Verifying Solutions
After simplifying an expression, it鈥檚 always a good practice to verify the solution. Verification ensures that the simplification was performed correctly. For example, having simplified \[\frac{32x^{5}-24x}{8}\] to 4\(x^5\) - 3\(x\), we verify by reversing the process. Multiply the simplified form by the common factor to see if we get the original numerator back.
Multiplying 8 by each term: 8\(4x^{5}-3x\) gives us 32\(x^{5}\) 鈥� 24\(x\). This matches the original numerator, which confirms that our simplification process was accurate. Regularly verifying solutions helps solidify one's understanding and avoid common mistakes.

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Most popular questions from this chapter

To prepare for Section 5.3, review combining like terms and evaluating expressions (Sections 1.6 and 1.8). Combine like terms. $$ 5 a-7 b-8 a+b $$

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