Unit 2 · Topic 1.8
Rational Functions and Zeros
Find zeros that belong to the function's domain, connect them with x-intercepts, and use numerator and denominator zeros to solve rational inequalities.
1. When Is a Fraction Equal to Zero?
A fraction equals zero precisely when its numerator is zero and its denominator is not zero.
\[R(x)=\frac{P(x)}{Q(x)}=0\quad\Longleftrightarrow\quad P(x)=0\text{ and }Q(x)\ne0.\]
The denominator condition is essential. Division by zero is undefined, so an excluded input can never be a zero of the rational function.
2. Zero, Root, and x-Intercept
If a real number \(a\) belongs to the domain and \(R(a)=0\), then:
- \(a\) is a zero of the function.
- \(a\) is a root, or solution, of \(R(x)=0\).
- The graph has an x-intercept at \((a,0)\).
These descriptions refer to the same real input, but only when the function is defined there.
3. A Reliable Zero-Finding Workflow
- Factor the numerator and denominator when possible.
- Record every zero of the original denominator as a domain restriction.
- Simplify common factors while keeping the original restrictions.
- Set the remaining numerator equal to zero.
- Reject any candidate excluded from the original domain.
- Report each accepted zero and its x-intercept.
A simplified formula may be algebraically equivalent on the original domain, but cancellation does not restore an excluded input.
4. Worked Example with a Common Factor
Analyze
\[R(x)=\frac{(x-3)(x+2)}{(x+2)(x-5)}.\]
| Step | Result | Meaning |
|---|
| Original denominator zeros | \(x=-2,5\) | Both values are excluded from the domain. |
| Simplified rule | \(R(x)=\frac{x-3}{x-5}\), \(x\ne-2,5\) | The restriction \(x=-2\) remains. |
| Remaining numerator zero | \(x=3\) | The denominator is nonzero, so this is a valid zero. |
| Graph feature | \((3,0)\) | The only x-intercept is at \(x=3\). |
The canceled value \(x=-2\) is not a zero because the original function is undefined there.
5. A Rational Function May Have No Real Zeros
Consider
\[A(x)=\frac{x^2+4}{x-1}.\]
The numerator equation \(x^2+4=0\) has no real solutions. Therefore \(A\) has no real zeros and its real graph has no x-intercepts.
Nonreal solutions of the numerator do not create x-intercepts on a real coordinate plane.
6. Multiplicity and Behavior at an x-Intercept
After common factors are handled, numerator multiplicity predicts local behavior at a valid zero.
| Numerator multiplicity | Sign near the zero | Graph behavior |
|---|
| Odd | Changes sign | Crosses the x-axis |
| Even | Keeps the same sign | Touches the x-axis and turns |
For \(B(x)=(x-2)^2/(x+1)\), \(x=2\) is a zero of multiplicity 2, so the graph touches the x-axis there without changing sign.
7. From Zeros to Rational Inequalities
To solve \(R(x)\ge0\) or \(R(x)\le0\), find every real zero of both the numerator and denominator. These critical values partition the number line into intervals.
- A numerator zero may be included for \(\ge\) or \(\le\).
- A numerator zero is excluded for \(>\) or \(<\).
- A denominator zero is always excluded.
- The sign is constant within each interval that contains no critical value.
8. Complete Sign-Chart Example
Solve
\[S(x)=\frac{(x+3)(x-1)}{(x+1)(x-4)}\le0.\]
The numerator zeros are \(-3\) and 1. The denominator zeros are \(-1\) and 4.
| Interval | Test value | Sign of \(S(x)\) | Selected? |
|---|
| \(( -\infty,-3)\) | \(-4\) | Positive | No |
| \((-3,-1)\) | \(-2\) | Negative | Yes |
| \((-1,1)\) | 0 | Positive | No |
| \((1,4)\) | 2 | Negative | Yes |
| \((4,\infty)\) | 5 | Positive | No |
Include the numerator zeros and exclude the denominator zeros:
\[\boxed{[-3,-1)\cup[1,4)}\]
9. Multiplicity Can Replace Repeated Testing
At a critical value with odd multiplicity, the sign changes. At one with even multiplicity, it stays the same. This applies to factors in either the numerator or denominator.
| Critical factor | Location | Multiplicity | Sign across the location |
|---|
| Numerator \((x-a)^3\) | Valid zero | Odd | Changes |
| Numerator \((x-a)^2\) | Valid zero | Even | Does not change |
| Denominator \((x-b)^3\) | Excluded input | Odd | Changes |
| Denominator \((x-b)^2\) | Excluded input | Even | Does not change |
Even when the sign does not change, mark the critical value because the function is zero or undefined there.
10. Comparing a Rational Function with a Nonzero Value
First move every term to one side and combine into one quotient. Solve
\[\frac{x+1}{x-2}\ge2.\]
\[\frac{x+1}{x-2}-2=\frac{x+1-2(x-2)}{x-2}=\frac{5-x}{x-2}\ge0.\]
The critical values are \(x=2\), where the expression is undefined, and \(x=5\), where it equals 0. Testing the three intervals gives
\[\boxed{(2,5]}\]
Do not cross-multiply an inequality by an expression of unknown sign; doing so may reverse the inequality on part of the domain.
11. Graphical and Numerical Evidence
- On a graph, zeros are x-intercepts where the function is defined.
- For \(R(x)>0\), select graph portions above the x-axis.
- For \(R(x)<0\), select graph portions below the x-axis.
- In a table, exact or approximate zeros appear where outputs equal 0 or change sign.
- A blank or undefined table entry is not a zero.
Graph and table evidence should agree with the factored expression and its domain.
12. Technology Workflow
- Factor or solve the numerator and denominator separately.
- Store all original denominator zeros as exclusions.
- Use a graphing tool to estimate x-intercepts and interval signs.
- Use a zero-finding command only on intervals where the graph is defined.
- Check every approximate zero in the original function.
- For inequalities, test one point in each interval and express the answer in interval notation.
13. Context and Domain Restrictions
A formula may have an algebraic domain broader than the meaningful contextual domain. For example, time, population, or length may require \(x\ge0\).
After solving an equation or inequality, intersect the mathematical solution with all contextual restrictions and state units when interpreting a zero.
14. Common Errors and AP Reasoning
- Accepting every numerator zero without checking the denominator.
- Restoring a canceled value to the domain.
- Calling a denominator zero an x-intercept.
- Including a denominator zero in an inclusive inequality.
- Excluding a valid numerator zero from \(\le\) or \(\ge\).
- Testing only the numerator's sign.
- Cross-multiplying by a variable expression without considering its sign.
- Using decimal graph estimates when exact factored values are available.
Complete response pattern: state the original restrictions, identify valid numerator zeros, partition the number line with every critical value, justify interval signs, and use correct endpoint inclusion.