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Chapter 5: Problem 18

Divide and check. $$\frac{4 x^{4} y-8 x^{6} y^{2}+12 x^{8} y^{6}}{4 x^{4} y}$$

Short Answer

Expert verified
1 - 2 x^{2} y + 3 x^{4} y^{5}

Step by step solution

01

- Break Down the Expression

Identify each term in the numerator: a) The first term is \(4 x^{4} y\). b) The second term is \(-8 x^{6} y^{2}\). c) The third term is \(12 x^{8} y^{6}\).
02

- Divide Each Term by the Denominator

Divide each term by the denominator \(4 x^{4} y\).a) For the first term: \(\frac{4 x^{4} y}{4 x^{4} y} = 1\). b) For the second term: \(\frac{-8 x^{6} y^{2}}{4 x^{4} y} = -2 x^{2} y\). c) For the third term: \(\frac{12 x^{8} y^{6}}{4 x^{4} y} = 3 x^{4} y^{5}\).
03

- Combine the Results

Combine the simplified terms to create the final simplified expression:\[1 - 2 x^{2} y + 3 x^{4} y^{5}\].
04

- Verify the Solution

To check the simplified expression, redistribute the denominator back by multiplying each term to see if it matches the original numerator.\( \left( 1 - 2 x^{2} y + 3 x^{4} y^{5} \right) (4 x^{4} y) = 4 x^{4} y - 8 x^{6} y^{2} + 12 x^{8} y^{6}\), which is indeed the original numerator, ensuring the simplification is correct.

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Key Concepts

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

Division of Polynomials
Understanding how to divide polynomials is an essential skill in algebra. In this exercise, we are dividing the polynomial \(\frac{4 x^{4} y-8 x^{6} y^{2}+12 x^{8} y^{6}}{4 x^{4} y}\).
We achieve this by breaking down the division into simpler steps.
  • First, identify and separate each term in the numerator.
  • Then, divide each term by the denominator individually.
For example, the first term \(4 x^{4} y\) divided by \(4 x^{4} y\) simplifies to 1. We repeat this for each term in the numerator. This process helps manage complexity, ensuring accuracy in polynomial division. It鈥檚 just like dividing regular numbers but with variables.
Simplifying Expressions
Once we have separated each term and divided it by the denominator, the next task is to simplify the results. Simplifying expressions involves reducing them to their most basic form.
  • For the term \(\frac{-8 x^{6} y^{2}}{4 x^{4} y}\), we simplify by dividing the coefficients and reducing the exponents separately.
  • This results in \(-2 x^{2} y\).
Similarly, divide \(12 x^{8} y^{6}\) by \(4 x^{4} y\) to get \(3 x^{4} y^{5}\).
Finally, combine all simplified terms to form the final expression: \(1 - 2 x^{2} y + 3 x^{4} y^{5}\). Simplifying expressions makes them easier to understand and work with.
Verifying Solutions
Verification is crucial in ensuring the accuracy of our mathematical solutions. After simplifying an expression, it's important to check that our work is correct.
We do this by redistributing the original denominator back through the simplified terms and seeing if we arrive back at the original numerator.
For example, multiplying each term in \(1 - 2 x^{2} y + 3 x^{4} y^{5}\) by \(4 x^{4} y\) should return us to \(4 x^{4} y - 8 x^{6} y^{2} + 12 x^{8} y^{6}\). By confirming this doing a reverse operation, we verify our solution is accurate. This practice of checking provides confidence and validation of our work.

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Most popular questions from this chapter

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{22} $$

Simplify. \(0.1 n^{2}+5-\left(-0.3 n^{2}+n-6\right)\)

Find a value of the variable that shows that the two expressions are not equivalent. Answers may vary. $$ \frac{t^{6}}{t^{2}} ; t^{3} $$

Explain in your own words why \(-5^{2} \neq(-5)^{2}\).

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=8 x^{3}+27, g(x)=2 x+3 $$
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