Polynomial and Rational Functions

This Grade 12 lesson extends the factoring, quadratic, domain, and transformation skills from Algebra and Functions. We use factors to connect formulas, zeros, signs, end behaviour, holes, and asymptotes.

Polynomial Structure and End Behaviour

A polynomial is a sum of terms axⁿ with whole-number exponents. Its degree is the greatest exponent, and its leading term controls the ends of the graph.

Degree and leading coefficient	Left end	Right end
Even, positive	Up	Up
Even, negative	Down	Down
Odd, positive	Down	Up
Odd, negative	Up	Down

Worked example: zeros, multiplicity, and signs

Analyse p(x) = (x + 2)(x − 1)².

Zeros: x = −2 and x = 1
Multiplicity at −2: 1, so the graph crosses the x-axis.
Multiplicity at 1:  2, so the graph touches and turns.

Leading term: x · x² = x³
Odd degree with positive leading coefficient:
left end down; right end up.

y-intercept: p(0) = (2)(−1)² = 2

A degree-three polynomial has at most three real zeros and at most two turning points. Multiplicity counts repeated zeros even when the graph has only two distinct intercepts.

Finding a Polynomial from Its Zeros

If r is a zero, then x − r is a factor. A point not on the x-axis determines the remaining scale factor.

Worked example: build the rule

A cubic has zeros −1, 2, and 4 and passes through (0, 8). Find its equation.

p(x) = a(x + 1)(x − 2)(x − 4)

Use (0, 8):
8 = a(1)(−2)(−4)
8 = 8a
a = 1

p(x) = (x + 1)(x − 2)(x − 4)

Rational Functions: Restrictions First

A rational function is a quotient of polynomials. Values that make the original denominator zero are excluded from the domain. Factor before simplifying so those restrictions are not lost.

Worked example: distinguish a hole from an asymptote

Analyse r(x) = (x² − 1)/(x² − x − 2).

r(x) = (x − 1)(x + 1) / [(x − 2)(x + 1)]
     = (x − 1)/(x − 2), but x ≠ −1 and x ≠ 2

x = −1 cancelled: a hole
hole y-value: (−1 − 1)/(−1 − 2) = 2/3
hole: (−1, 2/3)

x = 2 did not cancel: vertical asymptote x = 2
equal numerator and denominator degrees: horizontal asymptote y = 1

Polynomial and Rational Inequalities

Factor, mark every zero and undefined value on a sign chart, then test one point in each interval. Never include a denominator zero.

Worked example: rational inequality

Solve (x + 1)/(x − 3) ≤ 0.

Critical values: x = −1 (zero), x = 3 (undefined)

Interval       test        sign
(−∞, −1)      x = −2      positive
(−1, 3)       x = 0       negative
(3, ∞)        x = 4       positive

Include −1 because equality is allowed.
Exclude 3 because division by zero is never allowed.
Solution: [−1, 3)

Practice Set

1. State the degree, leading coefficient, and end behaviour of f(x) = −3x⁴ + 2x − 7.
2. For g(x) = (x − 5)²(x + 1), list the zeros and say whether the graph crosses or touches at each.
3. Find a quadratic with zeros 2 and −3 that passes through (0, −12).
4. Identify every restriction, hole, and vertical asymptote of (x² − 9)/[(x − 3)(x − 1)].
5. Solve (x − 2)/(x + 4) > 0.

Answer Checks

1. Degree 4, leading coefficient −3; both ends point down.
2. x = 5 has multiplicity 2, so it touches; x = −1 has multiplicity 1, so it crosses.
3. q(x) = a(x − 2)(x + 3). Since −12 = a(−2)(3), a = 2, so q(x) = 2(x − 2)(x + 3).
4. Restrictions: x ≠ 3, 1. Cancelling x − 3 gives a hole at (3, 3); x = 1 is a vertical asymptote.
5. Critical values are −4 and 2. The quotient is positive on (−∞, −4) and (2, ∞). Both endpoints are excluded.