Polynomial functions: degree, leading term, and intercepts

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

Try it

Summary

• Degree = highest exponent; leading coefficient = coefficient of that term.
• y-intercept is \(f(0)\); x-intercepts come from solving \(f(x)=0\).

Factoring to find zeros (roots) and using multiplicity

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

Try it

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.

Leading term test: end behavior of polynomials

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

Try it

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 functions: domain, holes, and vertical asymptotes

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

Try it

Summary

• Domain excludes denominator zeros.
• Canceled factor \(\Rightarrow\) hole; non-canceled denominator factor \(\Rightarrow\) vertical asymptote.

Horizontal asymptotes (and when slant asymptotes happen)

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

Try it

Summary

• \(\deg(\text{top})<\deg(\text{bottom}) \Rightarrow y=0\).
• \(\deg(\text{top})=\deg(\text{bottom}) \Rightarrow y=\dfrac{\text{leading coeff (top)}}{\text{leading coeff (bottom)}}\).
• \(\deg(\text{top})>\deg(\text{bottom}) \Rightarrow\) no horizontal asymptote; consider long division for slant when the difference is \(1\).

Solve rational equations (and avoid extraneous solutions)

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

Try it

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.

A fast checklist for polynomial & rational functions

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

Try it

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.