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Polynomials are some of our favorite functions.

The functions you are most familiar with are probably polynomial functions.

### What are polynomial functions?

Which of the following are polynomial functions?
$f(x) = 0$ $f(x) = -9$ $f(x) = 3x+1$ $f(x) = x^{1/2}-x +8$ $f(x) = -4x^{-3}+5x^{-1}+7-18x^2$ $f(x) = (x+1)(x-1)+e^x - e^x$ $f(x) = \frac {x^2 - 3x + 2}{x-2}$ $f(x) = x^7-32x^6-\pi x^3+45/84$

The phrase above “in the variable $x$” can actually change. is a polynomial in $y$, and is a polynomial in $\sin (x)$.

### Multiplying

Multiplying polynomials is based on the familiar property of arithmetic, distribution: $\displaystyle a\cdot ( b + c ) = ab + ac$.

The result is that we have multiplied every term of the first polynomial by every term of the second, then added the results together. In the case of two binomials (polynomials with only two terms), this is frequently referred to as FOIL.

There are several product formulas that arise repeatedly when working with binomials. You will likely have seen most of these before.

If $f(x)=4x^2-3x+1$, find $f(x+h)-f(x)$.
$h$ $4h^2-3h+1$ $8xh + 4h^2 - 3h$ $4x^2+8xh+4h^2-3x-3h+1$

### Factoring

Factoring is a bit like an inverse operation of multiplying polynomials. We start with the multiplied out polynomial, and ask what the individual factors were.

The easiest factors to deal with are common factors.

For trinomials (polynomials with three terms) of the form $ax^2 + bx + c$, we try to factor as $(px+r)(qx+s)$. We’ll start with an example with $a=1$.

The process is slightly more complicated when $a \neq 1$.

Remember, a root of a polynomial function is an $x$-value where the polynomial is zero. There is a close relationship between roots of a polynomial and factors of that polynomial. Precisely:

### Equations

A quadratic equation in $x$ is an equation which is equivalent to one with the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, with $a \neq 0$.

There are three major techniques you are probably familiar with for solving quadratic equations:

• Factoring.
• Completing the Square.

Each of these methods are important. Factoring is vital, because it is a valid approach to solve nearly any type of equation. Completing the Square is a technique that becomes useful when we need to rewrite certain types of expressions. The Quadratic Formula will always work, but has some limitations.

If you had used the quadratic formula, $\displaystyle x = \dfrac {-b \pm \sqrt {b^2-4ac}}{2a}$, instead of factoring or completing the square above, you would have found the same solutions.
Solve the quadratic equation $\displaystyle x^2 + 4 = 4\left (x+2\right )$.
$x = \pm 2$ $x=2\pm 2\sqrt {2}$ $x = -2, -4$ none of the above

Building on these methods, we’ll consider more general types of equations. A polynomial equation is an equation which is equivalent to one of the form $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0$.

Factoring is a process that helps us solve more than just quadratic equations, as long as we first get one side of the equation equal to zero.

The next theorem above is a deep fact of mathematics. The great mathematician Gauss proved the theorem in 1799.

Recall that the multiplicity of a root indicates how many times that particular root is repeated. The Fundamental Theorem of Algebra tells us that a polynomial equation of degree $n$, will have exactly $n$ complex solutions, once multiplicity is taken into account. As non-real solutions appear in complex-conjugate pairs (if our equation has real coefficients), we will always have an even number of non-real solutions. That means, taking multiplicity into account, a quadratic equation will have either $2$ or $0$ real solutions. A cubic equation will have either $3$ or $1$ real solution.

The equation $x^3 = 3x-2$ has $x=1$ as one solution. Find another solution.