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Chapter 8: Problem 10

Divide each of the following. Use the long division process where necessary. $$\frac{9 x-x^{2}}{x}$$

Short Answer

Expert verified
9 - x

Step by step solution

01

Set Up the Division

Rewrite the expression \(\frac{9x - x^2}{x}\) as \(9x - x^2 \, \div \, x\). This helps visualize the division process.
02

Simplify Each Term Individually

Divide each term of the numerator by the denominator individually. Start with the first term: \( \frac{9x}{x} \).
03

Simplify \( \frac{9x}{x} \)

Reduce the fraction \( \frac{9x}{x} \). Since \( x \) in the numerator and denominator cancel each other out, you're left with \( 9 \).
04

Simplify \( \frac{-x^2}{x} \)

Next, divide the second term: \( \frac{-x^2}{x} \). This simplifies to \( -x \) because one \( x \) cancels out from \( x^2 \). This leaves us with \( -x \).
05

Combine the Results

Combine the simplified terms from steps 3 and 4 to get the final result: \( 9 - x \).

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Key Concepts

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

simplifying algebraic expressions
In algebra, simplifying expressions is a crucial skill. It allows us to make complicated problems more manageable.
To simplify an algebraic expression, follow these steps:
  • Look for like terms. These are terms that have the same variables raised to the same power.
  • Combine the like terms by adding or subtracting their coefficients.
  • Apply the distributive property if necessary, which means multiplying across parenthesis.
  • Simplify any fractions by canceling common factors.
For example, in the exercise where we have \(9x - x^2 \div x\), we can simplify each fraction individually by eliminating the common variable, \(x\). Simplifying helps to see the underlying structure of the problem more clearly.
division of polynomials
Dividing polynomials might initially seem challenging, but long division makes it more systematic and easier to handle.
Follow these steps:
  • Write the polynomials in standard form, from highest power to lowest power.
  • Set up the long division by writing the divisor outside the division bracket and the dividend inside.
  • Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
  • Multiply the entire divisor by this first term of the quotient and subtract the result from the dividend.
  • Repeat this process with the new polynomial obtained until the remainder is zero or has a lower degree than the divisor.
In our example, we transform \(9x - x^2 \div x\) into individual divisions: \( \frac{9x}{x} - \frac{-x^2}{x} \), simplifying to 9 - x. This step-by-step approach ensures accuracy and helps break down complex polynomial expressions into simpler parts.
basic algebra techniques
Mastering basic algebra techniques is essential for solving more complex math problems. Here are some fundamental techniques to keep in mind:
  • **Distributive Property:** This helps in simplifying expressions by distributing a factor across terms inside parentheses, like \(a(b + c) = ab + ac\).
  • **Combining Like Terms:** Group terms with identical variables and powers, and then add or subtract their coefficients.
  • **Factoring:** Express a polynomial as a product of its factors. For example, factorizing \(x^2 - 9\) to \((x-3)(x+3)\).
  • **Transposition:** Moving a term from one side of an equation to another by performing the inverse operation to isolate variables.
In our exercise, we used several of these techniques. We simplified terms by canceling common variables and combined the simplified terms to reach our final answer. These foundational skills make solving more advanced algebra problems possible and less daunting.

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Most popular questions from this chapter

Use scientific notation to compute each of the following. $$\frac{(2,400)(0.003)}{(0.02)(0.004)}$$

Factor each of the following expressions as completely as possible. If an expression is not factorable, say so. $$c^{2}-6 c+5$$

Factor each of the following as completely as possible. If the expression is not factorable, say so. Try factoring by grouping where it might help. $$6 u^{3} v^{2}+18 u^{3} v^{3}-12 u^{3} v^{5}$$

Factor each of the following expressions as completely as possible. If an expression is not factorable, say so. $$x^{2}-x+20$$

Simplify the expression and express your final answer with positive exponents only. $$\frac{\left(x^{-2} y^{3}\right)^{-1}}{\left(x^{3}\right)^{-2}}$$
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