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Chapter 13: Problem 6

Find the zeros. $$ f(x)=\frac{10 x+4}{7 x-10} $$

Short Answer

Expert verified
Question: Find the zeros for the given function $$f(x)=\frac{10 x+4}{7 x-10}$$. Answer: The zero for the given function $$f(x)$$ is $$x = \frac{-2}{5}$$.

Step by step solution

01

Set the numerator equal to zero.

To find the zeros of the function, we need to find the x-values that make the numerator of the fraction zero. We set the numerator, which is $$10x + 4$$, equal to zero and solve for x: $$ 10x + 4 = 0 $$
02

Solve for x.

To solve for x, we subtract 4 from both sides and divide by 10: $$ 10x = -4 \\ x = \frac{-4}{10} $$
03

Simplify the result.

Now, we simplify the fraction: $$ x = \frac{-2}{5} $$
04

Check the denominator.

We need to make sure that the denominator is not equal to zero for the value of x found in the previous step. Substitute the value of x into the denominator: $$ 7x - 10 \\ 7\left(\frac{-2}{5}\right) - 10 $$ The result is not equal to zero, so we can conclude that the function has a zero at $$x = \frac{-2}{5}$$. Therefore, the zeros for the given function $$f(x)$$ are: $$ x = \frac{-2}{5} $$.

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Key Concepts

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

Numerator and Denominator
In rational functions like \( f(x) = \frac{10x + 4}{7x - 10} \), understanding the numerator and denominator is crucial. To find the zeros of a rational function, we only focus on the numerator, not the denominator. This is because a zero of a function is where the function equals zero. Hence, the fraction has to be zero, and the only way for this to happen is when the numerator is zero.

The **numerator**, in this context, is the part \( 10x + 4 \). On the other hand, the **denominator** is \( 7x - 10 \), which we are ensuring never equals zero, as that would make the entire expression undefined.

  • To find zeros, set the numerator equal to zero and solve for \( x \).
  • The denominator must be checked to ensure it doesn't become zero to prevent undefined expressions.
Solving Linear Equations
Solving linear equations is a fundamental skill when finding zeros of functions like this one. After setting the numerator equal to zero (\( 10x + 4 = 0 \)), our goal is to find \( x \).

Here's how you solve the linear equation:
  • First, isolate terms involving \( x \) on one side. Subtract 4 from both sides: \( 10x = -4 \).
  • Next, divide each term by 10 to isolate \( x \), leading to \( x = \frac{-4}{10} \).

Once you find \( x \), carefully verify that this solution doesn't make the denominator zero. If it does not, you've correctly found a zero of the function.
Simplifying Fractions
Once a fraction has been formed, such as our solution \( x = \frac{-4}{10} \), the next step is to simplify it. Simplifying fractions involves reducing them to their simplest form by removing the greatest common divisor of the numerator and the denominator.

For \( \frac{-4}{10} \), recognize that both 4 and 10 can be divided by their greatest common divisor, which is 2.
  • Divide the numerator by 2: \( -4 \div 2 = -2 \).
  • Divide the denominator by 2: \( 10 \div 2 = 5 \).

Thus, \( \frac{-4}{10} \) reduces to \( \frac{-2}{5} \). Simplifying fractions makes results clearer and helps in further calculations or interpretations. Always check if the fraction can be reduced further for the simplest expression possible.

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Most popular questions from this chapter

Divide common factors from the numerator and denominator in the rational expressions . Are the resulting expressions equivalent to the original expressions? $$ \frac{x^{4}}{x^{3}-x^{2}} $$

For the rational functions in Problems \(32-35,\) find all zeros and vertical asymptotes and describe the long-run behavior, then graph the function. $$ y=\frac{x^{2}-4}{x-9} $$

A rental car is driven \(d\) miles per day. The \(\operatorname{cost} c\) is \(c=\frac{2500+8 d}{d}\) cents per mile. (a) Find the cost per mile for a 100 mile per day trip. (b) How many miles must be driven per day to bring the cost down to 13 cents or less per mile? (c) Rewrite the formula for \(c\) to make it clear that \(c\) decreases as \(d\) increases. What is the limiting value of \(c\) as \(d\) gets very large?

Use the Remainder Theorem to decide whether the first polynomial is a factor of the second. If so, what is the other factor? $$ x-4,3 x^{3}-11 x^{2}-10 x+24 $$

For the rational functions in Problems \(32-35,\) find all zeros and vertical asymptotes and describe the long-run behavior, then graph the function. $$ y=\frac{x+3}{x+5} $$
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