MHF4U Polynomial Functions: Factor Theorem, Long Division & Inequalities Guide

In Ontario's MHF4U Advanced Functions, the Factor Theorem states that (x - b) is a factor of a polynomial P(x) if and only if P(b) = 0. Students combine the Rational Root Theorem with polynomial long division or synthetic division to factor degree 3+ polynomials and solve factorable polynomial inequalities using sign charts.

Official Ontario Curriculum Breakdown: MHF4U

• Ministry Course Code: MHF4U (Advanced Functions, Grade 12, University Preparation)
• Curriculum Strand: Polynomial and Rational Functions (B1: Degree and characteristics, B2: Factor theorem and inequalities)
• Prerequisite Course: MCR3U (Grade 11 Functions) or MCT4C
• Typical Semester: Taken in Semester 1 or Semester 2 prior to or concurrently with MCV4U
• University Weight: Core component of Top 6 Grade 12 U/M average for Engineering, Science, Commerce, and Computer Science

Worked Example 1: Factoring a Cubic Polynomial Completely Using the Factor Theorem

Problem: Factor the polynomial P(x) = 2x³ - 3x² - 11x + 6 completely, and state all real roots.

• Step 1 (Test rational root candidates): By the Rational Root Theorem, test factors of constant 6 divided by factors of leading coefficient 2: ±1, ±2, ±3, ±6, ±1/2, ±3/2.
• Step 2 (Apply Factor Theorem): Test P(1) = 2(1) - 3(1) - 11(1) + 6 = -6 ≠ 0. Test P(-1) = 2(-1) - 3(1) - 11(-1) + 6 = 12 ≠ 0. Test P(3) = 2(27) - 3(9) - 11(3) + 6 = 54 - 27 - 33 + 6 = 0! Therefore, (x - 3) is a linear factor.
• Step 3 (Polynomial Long Division / Synthetic Division): Divide (2x³ - 3x² - 11x + 6) by (x - 3): Quotient is 2x² + 3x - 2.
• Step 4 (Factor the quadratic quotient): 2x² + 3x - 2 factors into (2x - 1)(x + 2).
• Step 5 (Fully factored form and roots): P(x) = (x - 3)(2x - 1)(x + 2). Real roots are x = 3, x = 1/2, x = -2.

Teacher's Strategy: Always double-check your quotient by multiplying back or substituting another test point. Finding the first integer root quickly saves up to 10 minutes on timed Grade 12 exams.

Worked Example 2: Solving a Polynomial Inequality Using an Interval Sign Chart

Problem: Solve the inequality 2x³ - 3x² - 11x + 6 ≤ 0.

• Step 1 (Use factored form): (x - 3)(2x - 1)(x + 2) ≤ 0. The critical numbers (zeros) are x = -2, x = 1/2, and x = 3.
• Step 2 (Establish intervals on the real number line): (-∞, -2), (-2, 1/2), (1/2, 3), and (3, ∞).
• Step 3 (Test signs in each interval): For x ∈ (-∞, -2) [e.g. x = -3]: (-) × (-) × (-) = Negative ≤ 0 (SATISFIES). For x ∈ (-2, 1/2) [e.g. x = 0]: (-) × (-) × (+) = Positive > 0 (DOES NOT SATISFY). For x ∈ (1/2, 3) [e.g. x = 1]: (-) × (+) × (+) = Negative ≤ 0 (SATISFIES). For x ∈ (3, ∞) [e.g. x = 4]: (+) × (+) × (+) = Positive > 0 (DOES NOT SATISFY).
• Step 4 (State final solution with inclusive brackets): x ∈ (-∞, -2] ∪ [1/2, 3] or in set notation: {x ∈ ℝ | x ≤ -2 or 1/2 ≤ x ≤ 3}.

Teacher's Strategy: Notice the inclusive inequality (≤). Zeros are included in the solution set. If the inequality were strictly (<), use round parentheses () and exclude the boundary zeros.

The 5 Most Common Mistakes Ontario Students Make in MHF4U

• Dividing both sides of a polynomial inequality by a variable expression containing x (this loses roots or flips signs erroneously).
• Arithmetic sign errors when subtracting polynomial terms during synthetic and long division.
• Forgetting to include zeros in the solution set when the inequality has a ≤ or ≥ sign.
• Failing to list fractional root candidates (±p/q) when the leading polynomial coefficient is not 1.
• Plotting interval sign charts without testing actual intermediate numerical values.

University Admissions & Prerequisite Cutoff Impact

MHF4U Advanced Functions is universally evaluated across all Ontario STEM, Engineering, and Commerce programs. University programs like Waterloo Computer Science and U of T Rotman Commerce view polynomial fluency as a direct indicator of whether an applicant can handle university-level calculus.

Diagnostic Readiness Check: Test Your Knowledge

• If P(-4) = 0, what is the corresponding binomial factor? — Answer: (x + 4).
• What is the remainder when P(x) = x⁴ - 2 is divided by (x - 1)? — Answer: P(1) = 1⁴ - 2 = -1.
• Can a degree 4 polynomial have exactly 5 real roots? — Answer: No, by the Fundamental Theorem of Algebra, a degree n polynomial has at most n complex roots.