Advanced Functions Worksheets

What it is

The Remainder Theorem says that dividing a polynomial f(x) by (x − a) leaves a remainder equal to f(a) — you can find it by evaluating, no long division needed. The Factor Theorem is the special case where that remainder is 0. End behaviour, even/odd classification, and rational-function asymptotes are three more ways to read a polynomial's shape without graphing it — each detailed in its own step below.

Why it matters

These are the algebra tools behind graphing and analyzing higher-degree functions, exactly the skills that carry straight into calculus, where knowing a function's behavior without graphing it saves real time.

How to do it

1. Remainder Theorem: to find the remainder of f(x) ÷ (x − a), just compute f(a).
2. Factor Theorem: (x − a) is a factor of f(x) exactly when f(a) = 0.
3. End behaviour: check whether the degree is even or odd, and whether the leading coefficient is positive or negative.
4. Even/odd classification: check whether EVERY term's degree is even (Even), every term's degree is odd (Odd), or the degrees are mixed (Neither) — not just the leading term.
5. Asymptotes: set the denominator to 0 for vertical asymptotes; compare numerator/denominator degrees for the horizontal asymptote.

Examples by level

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

• Confusing the Remainder Theorem (evaluate f(a)) with the Factor Theorem (check if f(a) = 0).
• Forgetting that end behaviour depends on BOTH the degree's parity (even/odd) and the leading coefficient's sign, not just one.
• Confusing 'even/odd DEGREE' (end behaviour's leading-term shortcut) with 'even/odd FUNCTION' (a symmetry classification checking EVERY term) — a function can have an even-degree leading term and still be neither even nor odd if a lower-degree term breaks the pattern.
• Setting the numerator equal to 0 to find a vertical asymptote instead of the denominator.
• Assuming every rational function has a horizontal asymptote at y = 0 — it depends on comparing the degrees of the numerator and denominator.

Tips

• The Remainder Theorem and Factor Theorem are the same calculation — the Factor Theorem just asks whether the answer happens to be 0.
• End behaviour only needs two facts about the polynomial: is the degree even or odd, and is the leading coefficient positive or negative.
• Even/odd classification checks EVERY term's degree, not just the leading one — don't reuse end behaviour's leading-term shortcut here.
• Vertical asymptotes come from the denominator; horizontal asymptotes come from comparing degrees.

Key vocabulary

Remainder Theorem

the remainder of f(x) ÷ (x − a) equals f(a)

Factor Theorem

(x − a) is a factor of f(x) exactly when f(a) = 0

end behaviour

what a function's output does as x approaches positive or negative infinity — depends only on the leading term's degree and sign

even function

f(−x) = f(x) for every x; every term has an even degree, and the graph is symmetric about the y-axis

odd function

f(−x) = −f(x) for every x; every term has an odd degree, and the graph is symmetric about the origin

vertical asymptote

a vertical line the graph approaches but never crosses, where the function is undefined

horizontal asymptote

a horizontal line the graph approaches as x gets very large in either direction

Frequently asked questions

What is the Remainder Theorem?

When a polynomial f(x) is divided by (x − a), the remainder equals f(a) — you don't need to do the division at all, just evaluate.

What is the Factor Theorem?

(x − a) is a factor of f(x) exactly when f(a) = 0 — it's the Remainder Theorem's special case where the remainder is zero.

How do you tell if a function is even, odd, or neither?

Even if every term has an even degree (f(−x) = f(x), the graph is symmetric about the y-axis). Odd if every term has an odd degree (f(−x) = −f(x), symmetric about the origin). Neither if the degrees are mixed — this is a different question from end behaviour, which only looks at the leading term.

How do you find a rational function's vertical asymptote?

Set the denominator equal to 0 and solve — that's where the function is undefined and the graph shoots off to infinity.

How do you find a rational function's horizontal asymptote?

Compare the degrees of the numerator and denominator. If the denominator's degree is higher, the asymptote is y = 0. If they're equal, it's the ratio of the leading coefficients.