2.12 Polynomial functionsIB Maths: Analysis and Approaches HL: Revision notes

A polynomial of degree $n$ is $p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0}$ with $a_{n} \ne 0$. Its zeros are the values of $x$ with $p(x) = 0$; they are the roots of $p(x) = 0$ and the $x$-intercepts of $y = p(x)$.

• A degree $n$ polynomial has at most $n$ real zeros and at most $n - 1$ turning points.
• The end behaviour follows the leading term $a_{n}x^{n}$: for a cubic with $a_{3} > 0$, $y \to \infty$ as $x \to \infty$ and $y \to -\infty$ as $x \to -\infty$.
• A single zero: the graph crosses the axis. A repeated (double) zero, from a factor $(x - r)^{2}$: the graph touches the axis there. A triple zero: the graph crosses with a point of inflection.
• The $y$-intercept is $a_{0}$.

Remainder theorem: when $p(x)$ is divided by $(x - a)$, the remainder is $p(a)$. When divided by $(ax - b)$ the remainder is $p\left(\frac{b}{a}\right)$.

Factor theorem: $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$.

Example: $p(x) = 2x^{3} + ax^{2} + bx - 6$ with factor $(x - 2)$ and remainder $-6$ on division by $(x + 1)$ gives $4a + 2b = -10$ and $a - b = 2$, so $a = -1$, $b = -3$.

1. Find one root by trying factors of $\frac{a_{0}}{a_{n}}$ (e.g. $h(3) = 0$).
2. Divide by $(x - r)$ — long division, synthetic division or comparing coefficients — to get a quadratic.
3. Factorise or solve the quadratic; if its discriminant is negative there are no more real roots.

$x^{3} + 7x^{2} + 10x - 120 = (x - 3)(x^{2} + 10x + 40)$, and $\Delta = -60 < 0$, so $x = 3$ is the only real root.

For $a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{0} = 0$ (formula booklet):

$\text{sum of roots} = -\frac{a_{n-1}}{a_{n}}, \qquad \text{product of roots} = \frac{(-1)^{n}a_{0}}{a_{n}}.$

For $3x^{4} - 6x^{3} + kx^{2} + 5x - 12 = 0$: sum $= 2$, product $= -4$, whatever the value of $k$.

These results count complex roots and repeated roots too. They let you find sums and products without solving, e.g. the volume ($\alpha\beta\gamma$) of a box whose edges are the roots of a cubic. If each root is multiplied by 2, the sum doubles and the product is multiplied by $2^{n}$.