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Chapter 5: Problem 8

Divide and check. $$\frac{50 x^{5}-7 x^{4}+x^{2}}{x}$$

Short Answer

Expert verified
The simplified expression is \( 50x^4 - 7x^3 + x \).

Step by step solution

01

- Divide Each Term by x

First, split the given expression into separate terms and divide each term individually by \(x\). The expression is \( \frac{50x^5}{x} - \frac{7x^4}{x} + \frac{x^2}{x} \).
02

- Simplify Each Term

Next, simplify each division. For \( \frac{50x^5}{x} \), subtract the exponents: \( 50x^{5-1} = 50x^4 \). For \( \frac{7x^4}{x} \), it becomes \( 7x^{4-1} = 7x^3 \). Lastly, \( \frac{x^2}{x} \) simplifies to \( x^{2-1} = x \).
03

- Combine Simplified Terms

Combine the simplified terms to get the final result. Thus, the expression simplifies to \( 50x^4 - 7x^3 + x \).
04

- Verify the Result

To verify the result, multiply \( 50x^4 - 7x^3 + x \) by \( x \) and check if the original expression is obtained. Multiplying term-by-term: \( (50x^4)x = 50x^5 \), \( (-7x^3)x = -7x^4 \), \( (x)x = x^2 \). Thus, \( 50x^5 - 7x^4 + x^2 \) matches the original expression, 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.

Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operators (like addition, subtraction, multiplication, and division). In our exercise, the given algebraic expression is \(50x^{5} - 7x^{4} + x^{2}\). These expressions can be simplified or manipulated using various rules and techniques. Dividing these expressions involves dealing with each term separately. It's important to understand each part of the expression, especially how variables and their exponents behave when subjected to operations.
Exponent Rules
Exponents indicate how many times a number (the base) is multiplied by itself. For example, in the term \(x^5\), the base is \(x\) and the exponent is 5, meaning \(x\) is multiplied by itself 5 times. The key exponent rules used in our problem include:
  • When multiplying same bases: \(x^m \times x^n = x^{m+n}\)
  • When dividing same bases: \(x^m / x^n = x^{m-n}\)
.

In the exercise, each term of the expression was divided by \(x\). For instance, \( \frac{50x^5}{x} \) simplifies to \(50x^{5-1} = 50x^4\). Similarly, \( \frac{7x^4}{x} = 7x^{4-1} = 7x^3\) and \( \frac{x^2}{x} = x^{2-1} = x\). Subtracting exponents is crucial when dividing these terms, as shown in each calculation.
Simplifying Terms
Simplifying terms means reducing expressions to their simplest form. This often involves combining like terms and using mathematical operations to make expressions more manageable. In our example, we started with a complex algebraic expression and broke it down step-by-step.

  • First, we divided each term by \(x\).
  • Next, we simplified each term using exponent rules.
  • Finally, we combined the simplified terms to form the final expression: \(50x^4 - 7x^3 + x\).


Verifying the simplified expression involves performing the reverse operations to make sure the original expression is obtained. Multiplying \(50x^4 - 7x^3 + x\) by \(x\) should give the initial terms to ensure the simplification was done correctly. This helps confirm the accuracy of your work and understanding.

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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. $$ 2^{31} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{a^{4}}{b^{3}}\right)^{5} $$

Review factoring expressions and solving equations. $$ x+5=0 $$

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=6 x^{2}-11 x-10, g(x)=3 x+2 $$

In computer science, \(1 \mathrm{KB}\) of memory refers to 1 kilobyte, or 1 \(\times 10^{3}\) bytes, of memory. This is really an approximation of 1 \(\times 2^{10}\) bytes (since computer memory uses powers of \(2)\). The TI- 84 Plus Silver Edition graphing calculator has \(1.5 \mathrm{MB}\) (megabytes) of FLASH ROM, where \(1 \mathrm{MB}\) is \(1000 \mathrm{KB}\). How many bytes of FLASH ROM does this calculator have?
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