Polynomial functions and graphs (QCE Math Methods Unit 1)

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

Once a factor $(x - a)$ is confirmed, divide $P(x)$ by it to obtain a quotient of degree one less. Long division works for any divisor, while synthetic division is a fast shortcut for linear divisors. The result expresses $P(x) = (x - a)Q(x)$, after which $Q(x)$ (a quadratic, for a cubic) is factored by inspection or the quadratic formula. Equating coefficients is an equivalent method: write the quotient with unknown coefficients, expand, and match each power of $x$.

End behaviour

Set by leading term $ax^n$:

• $n$ even, $a > 0$: both ends $+\infty$.
• $n$ even, $a < 0$: both ends $-\infty$.
• $n$ odd, $a > 0$: left $-\infty$, right $+\infty$.
• $n$ odd, $a < 0$: left $+\infty$, right $-\infty$.

Root multiplicities

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

• $k = 1$: crosses the $x$-axis.
• $k = 2$: touches and turns (double root).
• $k = 3$: 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.

Turning points and shape

A degree-$n$ polynomial has at most $n - 1$ turning points, so a cubic has at most two and a quartic at most three. Between consecutive roots the graph must turn at least once, which together with the end behaviour and the crossing or touching behaviour at each root is usually enough to produce a correct shape without calculus. Symmetry can help too: an even polynomial (only even powers) is symmetric about the $y$-axis, and an odd polynomial (only odd powers) has rotational symmetry about the origin.

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.