MHF4U Polynomial Functions Practice Test With Worked Solutions

A polynomial question can ask you to connect several ideas: factors reveal zeros, multiplicity helps predict behaviour at an intercept, and the leading term describes the ends of the graph. This original MHF4U practice test brings those ideas together in ten questions with worked solutions.

It covers selected polynomial skills, not the entire Advanced Functions course. It is not an official examination. Try each question on paper before using graphing technology to check your work.

A short polynomial checklist

• Write the polynomial in descending powers when identifying its degree.
• Set each factor equal to zero to find the zeros of a factored polynomial.
• A zero of odd multiplicity crosses the x-axis; a zero of even multiplicity touches the axis and turns, for real polynomial graphs.
• Use the degree and leading coefficient for end behaviour.
• Check your factorization by expanding or substituting a convenient value.

For course context, see our MHF4U Advanced Functions guide.

Ten practice questions

• 1. State the degree, leading coefficient and end behaviour of f(x) = −2x⁵ + 3x² − 7.
• 2. For g(x) = (x + 2)²(x − 1), find the zeros, their multiplicities and the y-intercept. State whether the graph crosses or touches the x-axis at each zero.
• 3. Factor p(x) = x³ − 4x² − x + 4 completely and solve p(x) = 0.
• 4. Find the remainder when P(x) = 2x³ − 3x² + 4x − 5 is divided by x − 2.
• 5. Find k if x + 1 is a factor of Q(x) = x³ + kx² − 4x − 4. Then factor Q(x) completely.
• 6. Find a polynomial of least possible degree with real zeros −2, 1 and 3, each of multiplicity one, that passes through (0, 12).
• 7. Describe the graph of h(x) = −2(x − 3)⁴ + 5 as a transformation of y = x⁴. Give its domain, range and maximum point.
• 8. Solve (x + 3)(x − 1)(x − 2) > 0.
• 9. A rectangular box has dimensions x, x + 1 and x + 2 centimetres, where x > 0. Its volume is 60 cm³. Find its dimensions.
• 10. Determine the real zeros, multiplicities, y-intercept and end behaviour of r(x) = (x − 2)²(x + 1)(x − 4). Use these to describe a rough sketch.

Worked solutions

1. Degree and end behaviour

The highest power is 5, so the degree is 5 and the leading coefficient is −2. The leading term −2x⁵ controls the ends of the graph. As x approaches positive infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches positive infinity. In a sketch, the left end rises and the right end falls.

2. Zeros, multiplicity and the y-intercept

The factor (x + 2)² gives x = −2, multiplicity 2. The graph touches the x-axis and turns there. The factor (x − 1) gives x = 1, multiplicity 1, where it crosses.

For the y-intercept, substitute x = 0: g(0) = (2)²(−1) = −4. The intercept is (0, −4). Notice that the sign inside a factor is opposite to the zero it produces.

3. Factoring by grouping

Group the terms: x²(x − 4) − 1(x − 4). Factor out the common binomial to obtain (x² − 1)(x − 4). Since x² − 1 is a difference of squares, the complete factorization is (x − 1)(x + 1)(x − 4).

Therefore, p(x) = 0 at x = −1, 1 and 4. The zero-product property applies because the factored expression equals zero.

4. The remainder theorem

Dividing by x − 2 means evaluating P(2): 2(8) − 3(4) + 4(2) − 5 = 16 − 12 + 8 − 5 = 7. The remainder is 7, so x − 2 is not a factor.

5. The factor theorem

If x + 1 is a factor, Q(−1) = 0. Substitution gives −1 + k + 4 − 4 = 0, so k = 1. Then Q(x) = x³ + x² − 4x − 4 = x²(x + 1) − 4(x + 1) = (x + 1)(x² − 4).

The complete factorization is (x + 1)(x − 2)(x + 2).

6. Building a polynomial from zeros

Start with f(x) = a(x + 2)(x − 1)(x − 3). Substitute (0, 12): 12 = a(2)(−1)(−3) = 6a, giving a = 2.

One required polynomial is f(x) = 2(x + 2)(x − 1)(x − 3). It has degree 3, the least degree possible for three distinct zeros. The point determines the vertical scale; the zeros alone do not determine a unique polynomial.

7. Transformations and range

Move y = x⁴ three units right, stretch vertically by a factor of 2, reflect across the x-axis, and move five units up. The domain is all real numbers. Because (x − 3)⁴ is non-negative, −2(x − 3)⁴ + 5 cannot exceed 5. The range is y ≤ 5, and the maximum point is (3, 5).

8. A polynomial inequality

The zeros −3, 1 and 2 divide the real line into four intervals. Test x = −4, 0, 1.5 and 3. The product signs are, respectively, negative, positive, negative and positive.

The solution is −3 < x < 1 or x > 2, written (−3, 1) ∪ (2, ∞). Do not include the zeros because the inequality is strict. Testing intervals is safer than assuming every inequality includes everything beyond its largest zero.

9. A volume model

Volume equals the product of the dimensions: x(x + 1)(x + 2) = 60. Expanding gives x³ + 3x² + 2x − 60 = 0. Testing x = 3 gives 27 + 27 + 6 − 60 = 0.

Factoring gives (x − 3)(x² + 6x + 20) = 0. The quadratic discriminant is 36 − 80 = −44, so it has no real zeros. The only real solution is x = 3. The dimensions are 3 cm, 4 cm and 5 cm, and their product is 60 cm³.

10. Combining graph features

The zeros are −1 and 4, each of multiplicity 1, and 2, of multiplicity 2. The graph crosses at −1 and 4 and touches and turns at 2. The y-intercept is r(0) = 4(1)(−4) = −16.

The degree is 4 and the leading coefficient is positive, so both ends rise. Between −1 and 4, the graph is below the x-axis except where it touches at x = 2. These features support a rough sketch; they do not locate every turning point exactly.

What to review next

If your zeros were incorrect, revisit factors and sign changes. If your graph had the wrong ends, separate the degree from the sign of the leading coefficient. If your inequality answer was wrong, build a sign chart and test one number in each interval.

Keep an error log with three columns: the step you missed, the corrected rule and a fresh example. Reworking a similar problem without looking at the answer is more informative than simply rereading a solution.

Get help with the step that keeps going wrong

Explore Grade 12 math tutoring or book a free one-hour assessment to identify the foundations you need to strengthen.

Curriculum reference: Ontario Grades 11–12 Mathematics, MHF4U. These are original practice questions, not an official test.