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Polynomial

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KILL Integrate whiteboard.rnote into these written notes

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i just accidentally overwrote the file TwT

Factorisation

Like the ever familiar natural numbers, polynomials can be factorised, and even have a notion of primality.

Composite quadratics can be written as a product of linear terms. E.g.,

\begin{matrix} x^{2} + x - 6 = (x - 2) (x + 3) \end{matrix}

We may try to understand factorisation by considering the application of binomial multiplication in reverse.

\begin{matrix} (s x + t) (u x + v) \\ = & Distributivity \\ (s x + t) u x + (s x + t) v \\ = & Distributivity \\ s x u x + t u x + s x d + t v \\ = & Commutativity \\ s u x^{2} + t u x + s v x + t v \end{matrix}

Set this "expanded" expression as equal to a quadratic in standard form:

\begin{matrix} a x^{2} + b x + c & = s u x^{2} + t u x + s v x + t v \\ = (s x + t) (u x + v) \end{matrix}

The requirements to find factors are now clear. To factorise a quadratic is to find solutions to the previous equation, $s, t, u, v$ , such that

$a = s u$
$b = t u + s v$
$c = t v$

Factorisation is an effective technique to solve polynomial equations of any degree.

Leading coefficient of 1

When the leading coefficient $a$ is 1, factorisation is greatly simplified. $π (a) = (1,1)$ , reducing our system of equations to

\begin{matrix} x^{2} + b x + c = (x + s) (x + t) \\ s t = c \\ s + t = b \end{matrix}

The original polynomial's two solutions will be $- s$ and $- t$ .

Example

Example: $x^{2} + 5 x + 6$ .

Choose $s = 2$ and $t = 3$ . Assert that $s t = 6$ and $s + t = 5$ .

\begin{matrix} x^{2} + 5 x + 6 = (x + 2) (x + 3) \end{matrix}

Sure enough, multiplying the factors back together yields the original expression.

\begin{matrix} (x + 2) (x + 3) & = (x + 2) x + (x + 2) 3 \\ = x^{2} + 2 x + 3 x + 6 \\ = x^{2} + 5 x + 6 \end{matrix}

Example

Example: $x^{2} + 8 x + 15$ .

Choose $s$ and $t$ :

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\begin{matrix} x^{2} + 8 x + 15 = (x + 3) (x + 5) \end{matrix}

Completing the square

Quadratic formula

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\begin{matrix} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{matrix}

Rational equations

A rational equation is simply an equation where both sides are rational.My textbook defines a rational equation to be "an equation that contains a rational expression." I don't think that's what they meant. If their definition was the correct one, any absurd equation one could conceive could be made rational by some trivial rearranging, right?

A solution to a rational equation that would cause any of the contained expressions to become undefined is called an extraneous solution. It is recommended to acknowledge extraneous solutions $c$ by clarifying $x \neq c$ beside the relevant equation.

Basic terms

Degree of a polynomial

A polynomial's degree refers to the highest power any term is raised to.

The least possible degree of a polynomial is easy to identify visually — it's one greater than the number of 'turning points.'

Leading coefficient

The coefficient of the term with the greatest power.

The fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Roots

A polynomial of degree $n$ with complex coefficients has at most $n$ roots.

Polynomial inequalities

Solving algebraically

Polynomial inequalities may be solved purely-algebraically by factorising and analysing the polarity of the product in relation to the polarities of each factor. E.g., for a quadratic polynomial (one with exactly two factors), one should note that the product will be geq 0 only if both factors are positive, or if both factors or negative. Symbolically, this is $x y \geq 0 ⟺ (x \geq 0 \land y \geq 0) \lor (x \leq 0 \land y \leq 0)$ .

Example

file:quadratic-equations-assets/polynomial-inequality-algebraically.png

Solving graphically

That said, it is far easier and less error-prone to solve polynomial inequalities graphically. Simply identify the roots, and plot the function. The intervals for which $f x \geq 0$ become clear as day.

Though primitive, a fundamentally-similar technique can be used in a graphing calculator's absence. After identifying the polynomial's roots, the intervals between each solution (roughly) partition the real number line into several regionsE.g., if solutions lay at $x = - 2$ and $x = 5$ , our regions are $(- \infty, - 2)$ , $(- 2, 5)$ , and $(5, \infty)$ . . Knowing that each region will either be entirely positive or entirely negative, an arbitrary 'tracer' value in each region for $x$ can be selected to determine the entire region's polarity.

Example

file:quadratic-equations-assets/polynomial-inequality-semigraphically.png

Finding horizontal and vertical asymptotes of rational functions

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If both polynomials are the same degree, divide the coefficients of the highest degree terms to get the horizontal asymptote.
If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote.
If the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote. There is a slant asymptote.