Sketch and analyse polynomial functions of degree 3 and 4, using factored form to read roots and multiplicities, and applying the factor and remainder theorems

What this dot point is asking

QCAA wants you to sketch polynomial functions of degree $3$ and $4$, use the factor and remainder theorems to find factors, and read intercepts, end behaviour and root multiplicities from the factored form.

Remainder theorem

If $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$.

Factor theorem

$(x - a)$ is a factor of $P(x)$ if and only if $P(a) = 0$.

Rational roots theorem

For a polynomial with integer coefficients, any rational root $p/q$ (in lowest terms) has $p$ dividing the constant term and $q$ dividing the leading coefficient.

Procedure to factor a cubic

1. Identify rational-root candidates.
2. Test by computing $P(a)$.
3. Once a root $a$ is found, divide $P$ by $(x - a)$.
4. Factor the resulting quadratic.

Polynomial division

Long division of $P(x)$ by $(x - a)$. Synthetic division for linear divisors is a shortcut.

End behaviour

Set by leading term $ax^n$:

• IMATH_18 even, $a > 0$: both ends $+\infty$.
• IMATH_21 even, $a < 0$: both ends $-\infty$.
• IMATH_24 odd, $a > 0$: left $-\infty$, right $+\infty$.
• IMATH_28 odd, $a < 0$: left $+\infty$, right $-\infty$.

Root multiplicities

For a factor $(x - p)^k$:

• IMATH_33 : crosses the $x$-axis.
• IMATH_35 : touches and turns (double root).
• IMATH_36 : crosses with a horizontal tangent (point of inflection on axis).

Sketching from factored form

1. Read roots and multiplicities from the factors.
2. Find $y$-intercept by substituting $x = 0$.
3. Determine end behaviour from leading term.
4. Sketch through the roots with the correct crossing/touching behaviour.

Worked example

Sketch $y = (x + 2)(x - 1)^2 (x - 3)$.

Quartic, leading $+1$. Roots: $-2$ (crosses), $1$ (touches), $3$ (crosses). $y$-intercept: $y(0) = (2)(1)(-3) = -6$. Both ends $+\infty$.

Sketch enters from top left, crosses at $-2$, dips below, touches axis at $1$, descends below, crosses at $3$, exits top right.

Common traps

Wrong $Z$ direction. Factor $(x + 2)$ corresponds to root $-2$, not $+2$.

Missing the constraints of rational-roots theorem. Trying $x = 5$ when $5$ does not divide the constant wastes time.

Forgetting multiplicities. A double root touches; a triple root has a horizontal tangent.

In one sentence

Polynomial functions of degree $3$ or $4$ are factored using the rational-roots theorem to find candidate roots, the factor theorem to verify ($(x - a)$ is a factor iff $P(a) = 0$), and polynomial division to extract the linear factor; sketches read roots and multiplicities from the factored form, $y$-intercept from $x = 0$, and end behaviour from the leading term.