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Chapter 6: Problem 102

Simplify. Assume that no denominator is \(0 .\) $$ \frac{x^{2 k}-9}{3+x^{k}} $$

Short Answer

Expert verified
The simplified expression is \( x^k - 3 \).

Step by step solution

01

Identify the Expression

We need to simplify the given expression: \( \frac{x^{2k} - 9}{3 + x^{k}} \). Notice that this is a fraction with a difference of squares in the numerator.
02

Factor the Difference of Squares

The numerator \( x^{2k} - 9 \) is a difference of squares, which can be factored as \((x^k)^2 - 3^2\). This follows the formula \( a^2 - b^2 = (a - b)(a + b) \). Therefore, we rewrite the numerator as \( (x^k + 3)(x^k - 3) \).
03

Cancel Out Common Factors

Now, the expression becomes:\( \frac{(x^k+3)(x^k-3)}{3+x^k} \). Notice that \( x^k + 3 \) in the numerator is exactly the same as \( 3 + x^k \) in the denominator, as they are numerically equivalent. Therefore, these terms cancel each other out.
04

Simplify the Expression

After canceling the common term, the expression simplifies to \( x^k - 3 \). This is the simplest form of the original expression.

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Key Concepts

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

Difference of Squares
In algebra, the difference of squares is a fundamental concept where you have two square terms separated by a minus sign. The general formula for this is expressed as \(a^2 - b^2\).

The beauty of this formula lies in its ability to be factored easily into \((a - b)(a + b)\). This is a powerful tool in simplifying algebraic expressions because factoring often reveals the hidden structure of the expression.
  • For instance, in our exercise, the expression \(x^{2k} - 9\) is indeed a difference of squares.
  • Here, \(x^{2k}\) is the square of \(x^k\), and \(9\) is the square of 3.
  • Utilizing the difference of squares formula, we factor it into \((x^k - 3)(x^k + 3)\).
Understanding and identifying difference of squares is key in simplifying problems, making them less daunting and more approachable.
Factoring
Factoring involves breaking down an expression into a product of simpler terms that, when multiplied together, give the original expression. It's like taking a complex object apart, so you understand its components better.

In our scenario, after recognizing the difference of squares, factoring the expression \(x^{2k} - 9\) is straightforward. We used:
  • The formula\((a^2 - b^2 = (a - b)(a + b))\) to break it down into two binomials: \((x^k - 3)(x^k + 3)\).
Factoring is a crucial step because it leads to simplification by removing common terms or factors.
Grasping how to factor effectively not only simplifies problems but also enhances problem-solving skills.
Canceling Common Factors
Canceling common factors is a technique that further simplifies expressions by removing equivalent terms in both the numerator and denominator. It's like trimming away unnecessary parts to see the essential core.

In our expression, once we factored the difference of squares, we ended up with:
  • \( \frac{(x^k+3)(x^k-3)}{3+x^k} \)
Here, \(x^k + 3\) in the numerator matches \(3 + x^k\) in the denominator. Remember that addition is commutative, so \(x^k + 3\) and \(3 + x^k\) are essentially the same.

Cancelling these common factors reduces our expression to \(x^k - 3\). This simplification makes the expression easier to manipulate and understand in further mathematical operations.

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Most popular questions from this chapter

In a division exercise, if the divisor is \(x-3,\) the division process can be stopped when the degree of the remainder is a. 1 b. 0 c. 2 d. 3

The horsepower that can be safely transmitted to a shaft varies jointly as the shaft's angular speed of rotation (in revolutions per minute) and the cube of its diameter. A 2 -inch shaft making 120 revolutions per minute safely transmits 40 horsepower. Find how much horsepower can be safely transmitted by a 3 -inch shaft making 80 revolutions per minute.

Complete the following table for the inverse variation \(y=\frac{k}{x}\) over each given value of \(k .\) Plot the points on a rectangular coordinate system. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & {\frac{1}{4}} & {\frac{1}{2}} & {1} & {2} & {4} \\ \hline y =\frac{k}{x} & {} & {} & {} & {} & {} \\ \hline \end{array} $$ $$ k=\frac{1}{2} $$

For each given \(f(x)\) and \(g(x),\) find \(\frac{f(x)}{g(x)} .\) Also find any \(x\) -values that are not in the domain of \(\frac{f(x)}{g(x)} .\) (Note: since \(g(x)\) is in the denominator, \(g(x)\) cannot be \(0 .\).) $$ f(x)=7 x^{4}-3 x^{2}+2 ; g(x)=x-2 $$

Determine whether each division problem is a candidate for the synthetic division process. $$ \left(x^{7}-2\right) \div\left(x^{5}+1\right) $$
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