2.3 Polynomial and Rational Functions

Why This Section Matters

Polynomial and rational functions anchor a large share of Praxis 5165 function items. ETS tests factoring, zeros, vertex analysis, end behavior, vertical asymptotes, horizontal asymptotes, and removable discontinuities (holes). Teaching-scenario stems frequently ask how to respond when a student graphs a simplified rational expression without marking an excluded value.

Polynomial Basics

A polynomial uses nonnegative integer exponents and finitely many terms. Key skills:

• Factoring to find zeros
• Vertex form for quadratics: f(x) = a(x - h)² + k
• End behavior from the leading term

Worked Example: Maximum of a Quadratic

Find the maximum of f(x) = -x² + 4x + 1.

The leading coefficient is negative, so the parabola opens downward and has a maximum at the vertex.

x = -b/(2a) = -4/(2(-1)) = 2

f(2) = -(4) + 8 + 1 = 5

The maximum value is 5 at x = 2. Praxis distractors often give 2 (the x-coordinate of the vertex) instead of 5 (the maximum value).

Zeros and Multiplicity

Solving x² - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0, giving zeros 2 and 3.

If (x - 3)² is a factor, x = 3 is a zero with multiplicity 2 — the graph touches the x-axis at 3 but does not cross it.

Rational Functions: Structure

A rational function has the form R(x) = p(x)/q(x) where q(x) ≠ 0.

Domain

Exclude any x that makes q(x) = 0. For (x + 1)/(x - 4), the domain is all real numbers except x = 4.

Vertical Asymptotes vs. Holes

Worked Example: Hole on a Line

Simplify (x² - 4)/(x - 2) for x ≠ 2.

Factor numerator: (x - 2)(x + 2)

Cancel (x - 2): x + 2, with x ≠ 2

The graph matches y = x + 2 with a hole at (2, 4) because the original expression is undefined at x = 2.

Best teacher response: "For all x ≠ 2, the expressions are equivalent, so the graph follows the line y = x + 2 with a missing point — a hole — at (2, 4)."

Worked Example: Asymptotes

For f(x) = (2x + 1)/(x - 3):

• Vertical asymptote: x = 3 (denominator zero, no cancellation)
• Horizontal asymptote: degrees are equal (both 1), so y = 2/1 = 2 (ratio of leading coefficients)

Polynomial Division (When It Appears)

Long or synthetic division can rewrite improper rationals, but Praxis 5165 emphasizes factoring and interpretation over lengthy division drills. Know that division can reveal hidden linear behavior after cancellation.

Modeling and Graph Features

Polynomial models may describe projectile motion (quadratic), revenue curves, or volume constraints. Rational models often appear in average-cost or concentration settings where a variable appears in the denominator.

When sketching mentally, combine:

• Intercepts from zeros of numerator (watch domain)
• Asymptotes from uncancelled denominator zeros
• Holes from cancelled factors

Common Praxis Traps

• Dropping x ≠ 2 after simplifying (x² - 4)/(x - 2)
• Calling x = 2 a vertical asymptote when it is a hole
• Reporting vertex x instead of maximum/minimum value
• Assuming a rational graph cannot be linear away from discontinuities

Section Takeaways

Factor first, track excluded inputs, and connect algebraic simplification to holes and asymptotes on the graph. Drill vertex-maximum and rational-interpretation items on /practice/praxis-math.

Factoring Strategies for Praxis Items

When a polynomial is not obviously factorable, try these moves in order:

1. Greatest common factor — factor out shared monomials first
2. Trinomials — look for two numbers that multiply to ac and add to b in ax² + bx + c
3. Difference of squares — a² - b² = (a - b)(a + b)
4. Sum/difference of cubes — a³ ± b³ patterns appear occasionally

Worked Example: Finding Zeros

Find all real zeros of f(x) = x³ - 4x.

Factor: x(x² - 4) = x(x - 2)(x + 2).

Zeros: x = 0, 2, -2. The graph crosses the x-axis at each simple zero.

Graph Sketch Reasoning Without a Calculator

For f(x) = (x - 1)/(x + 2):

• Vertical asymptote at x = -2
• Horizontal asymptote y = 1 (equal degrees)
• x-intercept where numerator is zero: x = 1
• y-intercept at f(0) = -½

Listing these four features often suffices to eliminate three of four graph choices on Praxis 5165.

Connecting to Student Misconceptions

When students cancel (x - 2) and claim the domain is all reals, remind them that the original denominator still excluded x = 2. Simplified algebra does not erase the excluded input — it reveals a hole.

Similarly, students sometimes plot a vertical asymptote at a hole location. Teach them to factor first: if cancellation occurs, mark a hole; if not, mark an asymptote.

Polynomial Modeling Snapshot

Praxis modeling items usually stay at degree 2 or 3 with clear vertex or intercept meaning.