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MHF4U Polynomial Functions
Grade 12 Unit 1 — practice questions with full solutions

Degree, end behaviour, turning points and zeros — reading a polynomial's shape from its equation and vice versa.

Free MHF4U polynomial functions practice for Grade 12 students in Ontario. These 5 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MHF4U unit test, catching up on a lesson, or reviewing before the final exam.
MHF4U · Grade 12 Polynomial Functions 4 core 1 challenge
1Core
-3-2-1123-6-4-2246xyy = f(x)
From the cubic graphed above: (a) state the zeros, (b) write a possible equation in factored form given the graph passes through (1, −3), (c) describe the end behaviour, and (d) state the maximum possible number of turning points for a cubic.
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  1. (a) The curve crosses the x-axis at −2, 0 and 2.
  2. (b) Factored form: y = a(x + 2)(x)(x − 2). Substitute (1, −3): −3 = a(3)(1)(−1) = −3a, so a = 1.
  3. That gives y = x(x+2)(x−2), which expands to x³ − 4x.
  4. (c) Odd degree with a positive leading coefficient: as x → −∞, y → −∞; as x → +∞, y → +∞.
  5. (d) A degree-n polynomial has at most n − 1 turning points, so a cubic has at most 2.
Final answer(a) −2, 0, 2   (b) y = x(x+2)(x−2)   (c) from Q3 to Q1   (d) 2
2Core
Determine the degree, leading coefficient and end behaviour of f(x) = −2x⁴ + 5x³ − x + 7.
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  1. The degree is the highest exponent, which is 4.
  2. The leading coefficient is the number attached to that term: −2.
  3. Even degree means both ends point the same way; a negative leading coefficient turns them both downward.
  4. So as x → ±∞, y → −∞.
Final answerdegree 4, leading coefficient −2, both ends fall (Q3 to Q4)
3Core
Sketch-describe y = (x − 1)²(x + 3): state each zero, its order, and how the curve behaves at each.
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  1. Set each factor to zero. From (x−1)²: x = 1, appearing twice, so order 2.
  2. From (x+3): x = −3, order 1.
  3. An even-order zero means the curve touches the axis and turns back without crossing.
  4. An odd-order zero means the curve passes straight through.
  5. So at x = 1 it bounces off the axis; at x = −3 it crosses.
Final answerx = 1 (order 2, touches and turns); x = −3 (order 1, crosses)
4Core
A polynomial function of degree 3 has zeros at −1, 2 and 5, and f(0) = 20. Find f(x) in factored form.
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  1. Write the factored form using the zeros: f(x) = a(x + 1)(x − 2)(x − 5).
  2. Substitute x = 0: f(0) = a(1)(−2)(−5) = 10a.
  3. Set equal to 20: 10a = 20, so a = 2.
Final answerf(x) = 2(x + 1)(x − 2)(x − 5)
5Challenge
Determine the finite differences for f(x) = 2x³ + x at x = 0, 1, 2, 3, 4, and explain how the third differences confirm the degree.
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  1. Evaluate: f(0) = 0, f(1) = 3, f(2) = 18, f(3) = 57, f(4) = 132.
  2. First differences: 3, 15, 39, 75.
  3. Second differences: 12, 24, 36.
  4. Third differences: 12, 12 — constant.
  5. For a degree-n polynomial the nth differences are constant and equal a × n! Here 2 × 3! = 2 × 6 = 12. ✔
Final answerThird differences are constant at 12 = 2 × 3!, confirming degree 3

MHF4U Polynomial Functions — common questions

Short answers to the things students ask most about this unit.

How do you determine end behaviour?

Look at the degree and leading coefficient. Even degree sends both ends the same way; odd degree sends them opposite ways; a negative leading coefficient flips both.

What does the order of a zero do to the graph?

An odd-order zero crosses the x-axis. An even-order zero touches the axis and turns back without crossing.

More MHF4U practice — grade 12 advanced functions

MHF4U Polynomial Equations & InequalitiesMHF4U Rational FunctionsMHF4U Exponential & Logarithmic FunctionsMHF4U Trigonometric FunctionsAll MHF4U units
← All assignmentsUnit 2: Polynomial Equations & Inequalities →
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