Polynomial and power functions: structure before sketching: VCE Mathematical Methods Units 3 & 4

Loav · VCE Mathematical Methods · Units 3 & 4

F03 · 2026 curriculum · Original teaching guide

1. Choose the form that exposes the feature

A polynomial combines non-negative integer powers of x with real coefficients. Its degree is the highest power with a non-zero coefficient. The expanded form identifies degree and leading coefficient; a factorised form identifies zeros and their multiplicities; a completed-square form identifies a quadratic’s vertex. These forms describe the same function, but they answer different questions efficiently. Rewriting is useful when it reveals structure, not merely because expansion is possible.

For a quadratic, a(x − h)² + k gives vertex (h, k) and axis x = h. If a is positive, k is the minimum on the full real domain; if a is negative, k is the maximum. On a restricted domain, check whether h is allowed. A vertex outside the domain cannot provide an attained extremum for that restricted function. Also distinguish zeros, which are input values, from intercepts, which are coordinate pairs.

The leading term controls polynomial behaviour for large |x|. An odd-degree polynomial has opposite end directions; an even-degree polynomial has matching end directions. A positive leading coefficient makes the right end rise, while a negative coefficient makes it fall. This describes end behaviour, not every turn between the ends. Do not infer the exact number of turning points from the degree alone.

2. Factors explain crossing, touching and sign

If P(x) = (x − r)Q(x), then P(r) = 0. The converse is the factor theorem: a zero at r means x − r is a factor. Repeated factors matter. Near a root of even multiplicity, the factor keeps the same sign on either side, so the graph touches the axis without crossing there. Near a root of odd multiplicity, its sign changes, so the graph crosses. Other factors must be non-zero at that root when making this local argument.

For a sign diagram, mark the distinct real roots on a number line, choose one test input in each interval, and determine the product’s sign. A repeated even factor does not change sign, whereas a simple factor does. This gives a reliable basis for polynomial inequalities and for checking a sketch. At a root the output is exactly zero, so include it for a non-strict inequality and exclude it for a strict one.

3. Division and remainders provide useful checks

Dividing P(x) by x − r gives P(x) = (x − r)Q(x) + R, where R is constant. Substituting x = r leaves R = P(r), the remainder theorem. If this remainder is zero, division produces a factorisation and reduces the degree. Check a long-division result by multiplying the divisor by the quotient and adding the remainder; this catches sign errors that can otherwise spread through later work.

A supplied root often gives the first step in solving a cubic. After dividing out the corresponding linear factor, solve the remaining quadratic by factorisation or the quadratic formula. Report real solutions when working over ℝ; a negative discriminant means the quadratic contributes no real roots. A calculator’s approximate root list is useful for checking, but it does not replace an exact factorisation when exact values are requested.

Guided quartic sketch: combine roots and shape

Consider q(x) = (x² − 4)² = (x − 2)²(x + 2)². The two zeros, −2 and 2, each have multiplicity two, so the graph touches the x-axis at both. The y-intercept is (0, 16). The leading term is x⁴, so both ends rise. Every output is non-negative and zero is attained, giving domain ℝ and range [0, ∞). Substituting −x leaves the rule unchanged, so the graph is symmetric about the y-axis.

To establish the central shape precisely, a differentiation check gives q′(x) = 4x(x² − 4). Its sign is negative, positive, negative and positive in the successive intervals separated by −2, 0 and 2. Thus the graph decreases to (−2, 0), rises to a local maximum at (0, 16), falls to (2, 0), then rises again. The calculus guides develop this sign method in detail. Here it confirms why a quartic sketch needs more than its end directions: the two touching roots and the central maximum produce the complete shape.

4. Power functions include more than polynomials

A power function has the form f(x) = xʳ, with an appropriate real domain. Positive integer powers are defined for all real inputs. Even powers have symmetry about the y-axis and non-negative outputs; odd powers have rotational symmetry about the origin and take every real output. Negative integer powers are reciprocals, so x = 0 is excluded. For instance x⁻¹ has range ℝ \ {0}, whereas x⁻² has range (0, ∞).

For a rational exponent m/n written in lowest terms with n positive, use the real-root interpretation. An even denominator requires a non-negative base; if the exponent is negative, zero must also be excluded. An odd denominator allows negative bases, with zero again excluded for a negative exponent. For example, x^(2/3) = (∛x)² is defined for every real x and is non-negative. In contrast, x^(1/2) requires x ≥ 0.

Writing the fraction in lowest terms avoids a misleading root interpretation. The exponent 2/6 equals 1/3, and the real cube-root function is defined at negative inputs. Technology may use a different convention for non-integer powers of negative numbers. When a real cube root is intended, enter a real-root command or equivalent expression and verify the device’s interpretation. A complex-valued display is not evidence that the real-root function lacks those inputs.

Build a power-function sketch from the domain, symmetry, simple points and limiting behaviour. At a finite endpoint, show whether it is included. For negative powers, show the excluded input and the relevant asymptotes. Transformations then move those features using the same coordinate mapping as for other families. Do not apply a generic “power graph” shape without first checking the exponent’s sign and parity structure.

5. Sums, differences and products preserve their input conditions

The sum (f + g)(x) = f(x) + g(x) and product (fg)(x) = f(x)g(x) require both functions to be defined at the same input. Their domains are therefore the intersection of the two original domains. For a graphical sum, add the two ordinates at each shared x-coordinate. For a product, sign matters: two negative outputs multiply to a positive output, and a zero output from either function produces zero if the other is defined.

The difference (f − g)(x) = f(x) − g(x) has the same shared-domain requirement. On a graph, subtract g’s ordinate from f’s ordinate at each common input. Order matters: g − f is the negative of f − g. For example, f(x) = x² and g(x) = 2x on ℝ give (f − g)(x) = x² − 2x, which equals zero at x = 0 and x = 2, the intersections of the original graphs.

Algebraic simplification does not restore inputs excluded by the original definitions. If f(x) = 1/(x − 2), x ≠ 2, and g(x) = x − 2 on ℝ, then their product equals 1 on ℝ \ {2}. It is not the constant function with domain ℝ. Keep the original restriction alongside the simplified rule; a graph needs an open point at (2, 1). This is especially important when a CAS cancels factors automatically.

Practise, then check

1. Complete the square for 2x² − 12x + 11. State its vertex and solve the equation 2x² − 12x + 11 = 0 exactly.
2. Show that x − 3 is a factor of P(x) = x³ − 4x² + x + 6 and find all real roots.
3. Find the remainder when x³ − x² − 8x + 12 is divided by x − 1. Give the quotient too.
4. For Q(x) = −(x + 1)²(x − 4), state its end behaviour and solve Q(x) > 0.
5. State the maximal real domain and range of f(x) = x^(−2/3), interpreting the power through real cube roots.
6. Solve [∛(x + 2)]² = 9, explaining why there are two answers.
7. Let f(x) = √(x + 1) on [−1, ∞) and g(x) = x − 2 on [0, 5]. Find the domain and rule of f + g, and the zeros of fg.

Exam habits and technology checks

Keep exact factors until the graph’s roots and signs are secure. State restrictions before cancellation and distinguish “no real solution” from “no solution of any kind”. When a question requests an exact answer, retain radicals or factorised forms instead of rounding prematurely. Use a second representation, such as substitution into an expanded rule, to check a result obtained from factors.

A plotted curve can miss a touching root because there is no sign change, and a numerical solver can miss repeated roots. A symbolic factorisation or the algebraic multiplicity provides stronger evidence. For rational powers at negative inputs, verify that technology is using the real-root convention before trusting its graph or solution list.