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Chapter 6: Problem 8

Divide and check. $$ \frac{30 y^{8}-15 y^{6}+40 y^{4}}{5 y^{4}} $$

Short Answer

Expert verified
The simplified expression is \(6 y^{4} - 3 y^{2} + 8\).

Step by step solution

01

Rewrite the Division as Separate Fractions

Rewrite the given expression by separating each term in the numerator with the denominator. This gives: \[ \frac{30 y^{8}}{5 y^{4}} - \frac{15 y^{6}}{5 y^{4}} + \frac{40 y^{4}}{5 y^{4}} \]
02

Simplify Each Fraction

Simplify each fraction separately by dividing the coefficients and subtracting the exponents of like bases using the quotient rule for exponents (\(a^{m} / a^{n} = a^{m-n}\)). This gives: \[ \frac{30 y^{8}}{5 y^{4}} = 6 y^{4}, \; \frac{15 y^{6}}{5 y^{4}} = 3 y^{2}, \; \frac{40 y^{4}}{5 y^{4}} = 8 \]
03

Combine the Simplified Terms

Combine the results of the simplified fractions to get the final answer: \[ 6 y^{4} - 3 y^{2} + 8 \]
04

Verify the Simplification

To verify the simplification, multiply each term in the final answer by the original denominator (\(5 y^{4}\)) and check that it matches the original expression: \[ 5 y^{4} (6 y^{4} - 3 y^{2} + 8) = 30 y^{8} - 15 y^{6} + 40 y^{4} \]

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Key Concepts

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

Quotient Rule for Exponents
The Quotient Rule for Exponents is a fundamental principle when it comes to polynomial division. It states that when dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator: \( \frac{a^{m}}{a^{n}} = a^{m-n} \). Here's a step-by-step breakdown:
  • Identify the bases: In our exercise, all bases are \(y\).
  • Subtract the exponents: For \( \frac{30y^8}{5y^4}\), you subtract 4 from 8: \( y^{8-4} = y^4 \).
  • Divide the coefficients: The coefficients 30 and 5 are divided separately: \( \frac{30}{5} = 6 \).
Using this rule simplifies polynomial division and makes the expressions easier to manage. It's essential for simplifications and solving algebraic problems.
Simplifying Polynomial Fractions
To simplify polynomial fractions, follow these key steps:
  • Separate the terms: Break down the original fraction into individual fractions for each term in the numerator.
  • Simplify each fraction: Apply the quotient rule for exponents to simplify each fraction individually.
For instance, from the given exercise:
  • Start with: \( \frac{30y^8 - 15y^6 + 40y^4}{5y^4} \)
  • Separate fractions: \( \frac{30y^8}{5y^4} - \frac{15y^6}{5y^4} + \frac{40y^4}{5y^4} \)
  • Simplify fractions:
    \( \frac{30y^8}{5y^4} = 6y^4 \)
    \( \frac{15y^6}{5y^4} = 3y^2 \)
    \( \frac{40y^4}{5y^4} = 8 \)
This results in the final simplified expression: \( 6y^4 - 3y^2 + 8 \). This method ensures each term is dealt with systematically, making the process easier and accurate.
Verifying Polynomial Simplifications
Verification is the final step to ensure our simplification is correct. For the given polynomial, after simplifying \( \frac{30y^8 - 15y^6 + 40y^4}{5y^4} \rightarrow 6y^4 - 3y^2 + 8 \), we need to multiply back to check:
  • Start with the simplified polynomial: \( 6y^4 - 3y^2 + 8 \).
  • Multiply each term by the original denominator:
    \( 5y^4 (6y^4) = 30y^8 \)
    \( 5y^4 (-3y^2) = -15y^6 \)
    \( 5y^4 (8) = 40y^4 \).
  • Combine the terms:
    \( 30y^8 - 15y^6 + 40y^4 \).
If the resulting expression matches the original polynomial, our simplification is verified. This process ensures accuracy and builds confidence in algebraic manipulations.
Always take this final step to check your work!

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