R-ATSPE-2

Other stuff which is really useful to know:

• Theorem (The Fundamental Theorem of Algebra): Every real polynomial equation of degree $n$ has exactly $n$ solutions (some of which may be repeated or complex)
  • Real polynomials means that the coefficients of each term is real
• OR
• Every real polynomial $P(x)$ of degree $n\geq{1}$ can be factorised as a product of $n$ linear factors $a(x-a_{1})(x-a_{2})\dots(x-a_{n})$ where the $a_{i}$ are the zeroes of $P(x)$

e.g.

$$\begin{align} &x^{4}-4x^{3}-17x^{2}+110-150 \\ &=(x+5)(x-3)(x-(3+i))(x-(3-i)) \end{align} $$

This is useful because it can tell you about the shapes of the graphs.

Suppose $P(x)$ is a degree 7 polynomial. What is the maximum number of stationary points the graph of $y=P(x)$ can have?

• Something something derivative of degree 7 is degree 6 $\implies$ at most 6 roots (stuff above) $\implies$ at most 6 stationary points

• Theorem (Important!)

  • If $z$ is a complex solution to a polynomial equation (with real coefficients), then so is $\overline{z}$.

What important fact does this imply about the graphs of odd degree polynomials?

• It will always have at least one real solution!!! (odd vs parity of the conjugate pairs)

  • It follows from this theorem that the number of complex solutions (with non-zero imaginary part, else it is a real solution $:($) is always even.

• It follows that odd-degree polynomials always intercept the x axis!!!!!!!!!!!!!!

• Theorem

  • Suppose that $(x-a)^{n}$ is a factor of $P(x)$. Then...
  • If $n$ is odd, the graph of $y=P(x)$ crosses the $x$-axis at $a$.
  • If $n$ is even, the graph of $y=P(x)$ 'touches' but does not cross the $x$-axis at $a$.

We did WACE 2017 CF Question 2.

a) Use remainder theorem (show $\implies$ provide evidence as to why, i.e. substitute and show it becomes 0) b) Complex conjugate c) Proof by inspection

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