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Chapter 2: Problem 3

Fill in the blanks. The key numbers of a rational expression are its ____________ and its _____________ ______________.

Short Answer

Expert verified
The key numbers of a rational expression are its zeros and its undefined points.

Step by step solution

01

Identify the Zeros

Firstly, the Zeros/roots of a rational expression which are the solutions to equation \(f(x) = 0\). This is done by setting the numerator equal to zero and then solving for x.
02

Identify the Undefined Points

Secondly, the undefined points in a rational expression. They are determined by setting the denominator equal to zero and solving for x. These points are where the expression is undefined because you can't divide by zero.

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Key Concepts

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

Zeros of Rational Expressions
In rational expressions, zeros refer to the values of variables that make the expression equal to zero. Simply put, these are the solutions of the equation formed by setting the numerator of the rational expression to zero. Identifying zeros is a key step in understanding the behavior of a rational expression.
To find these zeros:
  • Set the numerator equal to zero. For instance, in an expression \( \frac{p(x)}{q(x)} \), only \( p(x) \) is considered at this step.
  • Solve the equation \( p(x) = 0 \) for the variable \( x \).
  • The solutions, or roots, are the zeros of the rational expression, where the expression itself equals zero.
Finding zeros is crucial as it tells us where the graph of the rational function intersects the x-axis. Remember, zeros occur only if the numerator equals zero while the denominator does not equal zero.
Undefined Points in Rational Expressions
Undefined points in rational expressions occur wherever the denominator equals zero. Understanding these points is essential for determining where a rational expression is undefined, as division by zero is not possible.
Here's how to find them:
  • Focus on the denominator \( q(x) \) of the expression \( \frac{p(x)}{q(x)} \).
  • Set \( q(x) \) equal to zero and solve for \( x \).
  • The solutions to \( q(x) = 0 \) are the points where the rational expression is undefined.
These undefined points are important because they often appear as vertical asymptotes on the graph of the rational expression. It's crucial to exclude these points when considering the domain of the expression.
Solving Rational Equations
Solving rational equations involves finding the values of the variables that satisfy the equality between two rational expressions. It requires understanding both the zeros and undefined points to ensure valid solutions.
Here鈥檚 a simplified process to solve such equations:
  • Firstly, identify and exclude any values of \( x \) that cause any denominator to be zero. These are not part of the solution set.
  • Multiply both sides of the equation by a common denominator to eliminate the fractions.
  • Solve the resulting polynomial equation for \( x \).
  • Check each solution by substituting back into the original equation to ensure that it doesn't make any denominator zero.
By following these steps, you ensure that the solution is valid within the context of rational equations. Not all potential solutions will necessarily fulfill the original equation without creating undefined expressions.

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{3}-x^{2}+x+39$$

Use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-1$$

The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b)\) (GRAPH CANNOT COPY) Match each complex number with its corresponding point. (i) 3 (ii) \(3 i\) (iii) \(4+2 i\) (iv) \(2-2 i\) (v) \(-3+3 i\) (vi) \(-1-4 i\)

Think About It \(\quad\) A cubic polynomial function \(f\) has real zeros \(-2, \frac{1}{2},\) and \(3,\) and its leading coefficient is negative. Write an equation for \(f\) and sketch its graph. How many different polynomial functions are possible for \(f ?\)
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