Polynomials

1. Chapter at a glance

• A polynomial of degree 1 is linear, degree 2 is quadratic and degree 3 is cubic.
• A real number k is a zero of p(x) if p(k) = 0.
• The zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
• The graph of a quadratic polynomial is a parabola opening upwards (a > 0) or downwards (a < 0).
• A quadratic polynomial has at most two zeroes; a cubic polynomial has at most three zeroes.
• For quadratic ax² + bx + c (a ≠ 0), sum of zeroes = –b/a and product of zeroes = c/a.
• For cubic ax³ + bx² + cx + d (a ≠ 0), sum of zeroes = –b/a, sum of products of zeroes taken two at a time = c/a, and product of zeroes = –d/a.
• A polynomial of degree n has at most n zeroes.

2. Definitions, theorems and results

• Linear polynomial: A polynomial of degree 1.
• Quadratic polynomial: A polynomial of degree 2; general form ax² + bx + c where a, b, c are real numbers and a ≠ 0.
• Cubic polynomial: A polynomial of degree 3; general form ax³ + bx² + cx + d where a, b, c, d are real numbers and a ≠ 0.
• Zero of a polynomial: A real number k is a zero of p(x) if p(k) = 0.
• Geometrical meaning (linear): The zero of ax + b (a ≠ 0) is the x-coordinate of the point where y = ax + b intersects the x-axis; exactly one zero.
• Geometrical meaning (quadratic): The zeroes are the x-coordinates of the points where the parabola y = ax² + bx + c intersects the x-axis. Three cases exist: two distinct zeroes (cuts x-axis at two points), one zero (touches x-axis at one point), or no zero (does not cut x-axis).
• Result on number of zeroes: A quadratic polynomial has at most two zeroes; a cubic polynomial has at most three zeroes. In general, a polynomial of degree n has at most n zeroes (graph intersects x-axis at most n times).
• Relationship for quadratic (derived by factorisation and coefficient comparison): If α and β are zeroes of ax² + bx + c, then α + β = –b/a and αβ = c/a.
• Relationship for cubic (verified by substitution and coefficient comparison): If α, β, γ are zeroes of ax³ + bx² + cx + d, then α + β + γ = –b/a, αβ + βγ + γα = c/a and αβγ = –d/a.
• No theorems are explicitly stated as requiring proof in the exam; the relationships are presented through verification and derivation.

3. Formula sheet

4. Solved-example patterns

• Type 1: Find number of zeroes from graph — Count the distinct intersection points of the curve y = p(x) with the x-axis.
• Type 2: Find zeroes of quadratic by factorisation and verify relationships — Factorise ax² + bx + c into linear factors; set each factor = 0 to obtain zeroes; compute sum and product directly and compare with –b/a and c/a.
• Type 3: Find zeroes of quadratic without obvious factors (e.g., difference of squares) — Rewrite using identities such as x² – k = (x – √k)(x + √k); obtain zeroes and verify sum/product.
• Type 4: Form quadratic polynomial given sum and product of zeroes — Assume form x² – (sum)x + product (or k times this); substitute given values of sum and product.
• Type 5: Verify zeroes and relationships for cubic — Substitute each given value into p(x) to confirm p(value) = 0; compute the three expressions (sum, sum of products two at a time, product) and compare with –b/a, c/a, –d/a.

5. Common mistakes and exam pitfalls

• Forgetting the negative sign when writing sum of zeroes = –b/a.
• Writing product as –c/a instead of c/a for quadratics.
• Assuming every quadratic must have two zeroes (missing the “at most” and the no-zero case).
• Sign errors when expanding or comparing coefficients for cubics (especially product = –d/a).
• Confusing “number of zeroes” with “number of distinct zeroes” when the graph touches the x-axis.
• Using incorrect coefficients when the leading coefficient a ≠ 1 (must divide correctly by a).