Learn Polynomial & Rational Functions: Interactive Problems and Worked Examples
Use the question set below to master polynomial functions and rational functions with the exact skills that show up in tests and homework: degree and leading coefficient, x-intercepts (real zeros / roots) and the factor theorem, multiplicity and how a graph crosses or touches the x-axis, end behavior using the leading term test, and rational-function essentials like domain restrictions, vertical asymptotes, holes (removable discontinuities), horizontal asymptotes and slant (oblique) asymptotes, intercepts, and solving rational equations with extraneous-solution checks. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Jawab rangkaian soal dan tinjau kesalahanmu di akhir.
How this polynomial and rational functions practice works
1. Take the practice set: answer the polynomial and rational functions questions below.
2. Open the lesson (optional): review zeros, factoring, intercepts, end behavior, domain, holes, and asymptotes with clear examples.
3. Retry: return to the question set and apply the polynomial and rational function rules immediately.
What you will learn in the polynomial & rational functions lesson
Polynomial function fundamentals
Degree, leading term, and leading coefficient
Intercepts: y-intercept \(f(0)\) and x-intercepts (real zeros)
End behavior from the leading term (even/odd degree, positive/negative leading coefficient)
Zeros, factors & multiplicity
Factoring patterns and the zero-product property
Multiplicity: when the graph crosses vs. touches the x-axis
Finding real zeros and writing polynomials in factored form
Purpose: Build a clear understanding of polynomial functions and rational functions so you can find intercepts, zeros / roots, and multiplicity, describe end behavior using the leading term, and analyze rational-function features like domain restrictions, holes (removable discontinuities), vertical asymptotes, and horizontal or slant (oblique) asymptotes. You will also practice solving rational equations while checking for extraneous solutions.
Success criteria
Identify a polynomial’s degree and leading coefficient.
Find y-intercepts by evaluating \(f(0)\) and interpret them correctly.
Find real zeros of polynomials using factoring and the zero-product property.
Use multiplicity to predict whether the graph crosses or touches the x-axis.
Describe polynomial end behavior using the leading term test.
Find the domain of a rational function by excluding denominator zeros.
Identify holes (canceled factors) and locate the hole’s coordinate.
Find vertical asymptotes from non-canceled denominator factors.
Determine horizontal asymptotes (and when degrees allow a slant asymptote).
Solve rational equations by clearing denominators and checking for extraneous solutions.
Key vocabulary
Polynomial function: \(p(x)=a_nx^n+\cdots+a_1x+a_0\) with nonnegative integer exponents.
Degree: the highest exponent with a nonzero coefficient.
Leading coefficient: the coefficient of the highest-degree term.
Zero / root: a value \(x=r\) where \(p(r)=0\); also an x-intercept when real.
Multiplicity: how many times a factor repeats, e.g., \((x-1)^2\) has multiplicity \(2\).
Rational function: \(f(x)=\dfrac{p(x)}{q(x)}\) where \(p,q\) are polynomials and \(q(x)≠ 0\).
Hole: a removable discontinuity from a canceled factor.
Vertical asymptote: where \(f(x)\) grows without bound near a denominator zero that does not cancel.
Horizontal asymptote: a line \(y=L\) describing end behavior as \(x\to\pm\infty\).
Slant (oblique) asymptote: a line asymptote from long division when degrees differ by \(1\).
Quick pre-check
Pre-check 1: What is the y-intercept of \(f(x)=2x^2-8x+3\)?
Hint: The y-intercept is \(f(0)\).
Pre-check 2: What is the horizontal asymptote of \(f(x)=\dfrac{4x+1}{2x-3}\)?
Hint: When degrees are equal, the horizontal asymptote is the ratio of leading coefficients.
Polynomial Basics
Polynomial functions: degree, leading term, and intercepts
Learning goal: Identify degree and leading coefficient, then find intercepts quickly and correctly.
Key idea
A polynomial function looks like \[ p(x)=a_nx^n+\cdots+a_1x+a_0, \] where the exponents are whole numbers and \(a_n≠ 0\). The degree is the highest exponent \(n\), and the leading coefficient is \(a_n\).
y-intercept: evaluate \(p(0)\).
x-intercepts: solve \(p(x)=0\). Real solutions are x-intercepts.
Worked example
Example: For \(p(x)=-2x^3+x\), find the degree and the y-intercept.
The highest power is \(3\), so the degree is \(3\). The y-intercept is \(p(0)\): \[ p(0)=-2(0)^3+0=0. \] So the y-intercept is \((0,0)\).
Try it
Try it 1: What are the degree and leading coefficient of \(p(x)=7x^4-2\)?
Hint: The leading term is the highest-degree term.
Try it 2: What is the y-intercept of \(p(x)=x^4-16\)?
Hint: The y-intercept is \(p(0)\).
Summary
Degree = highest exponent; leading coefficient = coefficient of that term.
y-intercept is \(f(0)\); x-intercepts come from solving \(f(x)=0\).
Zeros & Multiplicity
Factoring to find zeros (roots) and using multiplicity
Learning goal: Factor polynomials to find real zeros, then use multiplicity to predict graph behavior at each zero.
Key idea
If a polynomial is written in factored form, you can use the zero-product property: \[ (x-a)(x-b)=0 \Rightarrow x=a \text{ or } x=b. \] A repeated factor gives multiplicity. If \((x-r)^m\) is a factor:
If \(m\) is odd, the graph typically crosses the x-axis at \(x=r\).
If \(m\) is even, the graph typically touches/bounces at \(x=r\).
Worked example
Example: Find the real zeros of \(p(x)=x^4-16\).
Factor as a difference of squares: \[ x^4-16=(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4). \] The factor \(x^2+4\) has no real zeros. So the real zeros are: \[ x=-2,\quad x=2. \]
Try it
Try it 1: For \(f(x)=(x-1)^2(x+3)\), what is the multiplicity of the zero at \(x=1\)?
Hint: Multiplicity is the exponent on the factor \((x-1)\).
Try it 2: What are the real zeros of \(p(x)=x^3-9x\)?
Hint: Factor \(x^3-9x=x(x^2-9)=x(x-3)(x+3)\).
Summary
Factor \(p(x)\) and solve each factor \(=0\) to find real zeros.
Multiplicity tells you whether the graph crosses (odd) or touches (even) the x-axis.
End Behavior
Leading term test: end behavior of polynomials
Learning goal: Use the degree (even/odd) and the sign of the leading coefficient to predict what happens as \(x\to\pm\infty\).
Key idea
For large \(|x|\), a polynomial behaves like its leading term \(a_nx^n\). This is the leading term test:
Even degree (\(n\) even): both ends go the same direction.
Odd degree (\(n\) odd): the ends go opposite directions.
Positive leading coefficient: right end goes up.
Negative leading coefficient: right end goes down.
Worked example
Example: What is the end behavior of \(f(x)=-2x^3+x\)?
The leading term is \(-2x^3\) (odd degree, negative leading coefficient). So: \[ x\to\infty \Rightarrow f(x)\to -\infty,\qquad x\to -\infty \Rightarrow f(x)\to \infty. \]
Try it
Try it 1: What is the end behavior of \(p(x)=5x^4-x^2\)?
Hint: Even degree with positive leading coefficient means both ends go up.
Try it 2: What is the end behavior of \(p(x)=-x^5+2x\)?
Hint: Odd degree with negative leading coefficient means right end down, left end up.
Summary
Use the leading term \(a_nx^n\) to predict end behavior.
Even degree: same direction; odd degree: opposite directions; sign of \(a_n\) sets the right end.
Rational Basics
Rational functions: domain, holes, and vertical asymptotes
Learning goal: Find domain restrictions and distinguish holes from vertical asymptotes using factoring and cancellation.
Key idea
A rational function is a quotient of polynomials: \[ f(x)=\frac{p(x)}{q(x)}. \] The domain excludes values where \(q(x)=0\). When you factor and simplify:
If a factor cancels, the original function has a hole at that \(x\)-value (removable discontinuity).
If a denominator factor does not cancel, it creates a vertical asymptote.
Worked example
Example: Where is the hole in \(f(x)=\dfrac{(x-2)(x+1)}{x-2}\)?
Cancel the common factor (but remember the restriction): \[ f(x)=\frac{(x-2)(x+1)}{x-2}=x+1,\quad x≠ 2. \] So there is a hole at \(x=2\). The y-value comes from the simplified function: \[ y=2+1=3. \] The hole is at \((2,3)\).
Try it
Try it 1: Where are the vertical asymptotes of \(f(x)=\dfrac{x^2+1}{x^2-4}\)?
Hint: Vertical asymptotes come from denominator zeros that do not cancel.
Try it 2: What is the domain of \(f(x)=\dfrac{1}{x-6}\)?
Horizontal asymptotes (and when slant asymptotes happen)
Learning goal: Determine horizontal asymptotes quickly using degree comparisons, and recognize when long division is needed.
Key idea
For \(f(x)=\dfrac{p(x)}{q(x)}\), compare degrees:
If \(\deg(p)<\deg(q)\), the horizontal asymptote is \(y=0\).
If \(\deg(p)=\deg(q)\), the horizontal asymptote is the ratio of leading coefficients.
If \(\deg(p)>\deg(q)\), there is no horizontal asymptote. If the degrees differ by \(1\), the graph may have a slant (oblique) asymptote from long division.
Worked example
Example: What is the horizontal asymptote of \(f(x)=\dfrac{5x^3+1}{x^3+4}\)?
The degrees are equal (\(3\) and \(3\)), so the horizontal asymptote is the ratio of leading coefficients: \[ y=\frac{5}{1}=5. \]
Try it
Try it 1: What is the horizontal asymptote of \(f(x)=\dfrac{7x^4-2}{2x^4+3}\)?
Hint: Degrees are equal, so use the ratio of leading coefficients.
Try it 2: What is the horizontal asymptote of \(f(x)=\dfrac{1}{x^2}\)?
Hint: Degree on top is less than degree on bottom, so \(y=0\).
Learning goal: Clear denominators using the LCD, solve, then check restrictions so you do not keep invalid answers.
Key idea
To solve a rational equation:
Step 1: Write domain restrictions (values that make a denominator zero).
Step 2: Multiply both sides by the LCD to clear fractions.
Step 3: Solve the resulting equation.
Step 4: Check solutions in the original equation to eliminate extraneous solutions.
Worked example
Example: Solve \(\dfrac{x-1}{x+2}=2\).
Restriction: \(x≠ -2\). Multiply both sides by \(x+2\): \[ x-1=2(x+2). \] \[ x-1=2x+4 \Rightarrow -5=x \Rightarrow x=-5. \] Check: \((-5)+2=-3≠ 0\), so \(x=-5\) is valid.
Try it
Try it 1: Solve \(\dfrac{x+1}{x-2}=3\).
Hint: Restriction \(x≠ 2\). Multiply: \(x+1=3(x-2)\), then solve.
Try it 2: Where is the vertical asymptote of \(f(x)=\dfrac{x+1}{x-2}\)?
Hint: Vertical asymptotes occur where the denominator is zero (and does not cancel).
Summary
Clear denominators with the LCD, solve, then check restrictions to remove extraneous solutions.
Never allow a denominator to be zero in your final answer.
Big Picture
A fast checklist for polynomial & rational functions
Learning goal: Combine the skills into a reliable checklist, then finish with a final check.
Graphing checklist (high-value steps)
1) Identify the function type: polynomial or rational.
2) Intercepts: compute \(f(0)\) for the y-intercept; solve \(f(x)=0\) for x-intercepts.
3) For polynomials: factor if possible, list zeros with multiplicity, and use end behavior from the leading term.
4) For rational functions: factor numerator/denominator, note domain restrictions, cancel common factors (holes), keep non-canceled denominator factors (vertical asymptotes).
5) End behavior: find horizontal or slant asymptotes from degree comparison (or long division).
6) Final check: make sure every restriction is respected (no denominator \(=0\)).
Worked example: a common hole pattern
Example: Where is the hole in \(f(x)=\dfrac{(x-4)(x+1)}{x-4}\)?
Cancel the common factor (but keep the restriction \(x≠ 4\)): \[ f(x)=x+1,\quad x≠ 4. \] The hole is at \(x=4\) and the y-value is \(4+1=5\). So the hole is \((4,5)\).
Try it
Try it 1: Where is the hole in \(f(x)=\dfrac{(x-4)(x+1)}{x-4}\)?
Hint: Cancel the common factor and evaluate the simplified function at the excluded x-value.
Try it 2: Determine the horizontal asymptote of \(y=\dfrac{5x^3-x+1}{2x^3+4}\).
Hint: Degrees are equal, so use the ratio of leading coefficients.
Final recap
Polynomials: degree + leading coefficient, intercepts, zeros by factoring, multiplicity, and end behavior from the leading term.
Rational functions: domain restrictions, holes (canceled factors), vertical asymptotes (non-canceled denominator factors), and end behavior via horizontal/slant asymptotes.
Rational equations: clear denominators with the LCD, solve, then check for extraneous solutions.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the polynomial or rational function skill you need.
