Polynomials

A polynomial (in one variable) is something of the form

$$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0.$$

The a's stand for numbers, and x is the variable (but you can use any letter for the variable). In other words, a polynomial is a sum of nonnegative powers of the variable, possibly multiplied by numbers. (The $a_0$ term is $a_0 \cdot
   x^0$ , so it fits this description.)


Example. Here are some polynomials:

$$x^3 + 4x - 5 \hskip0.5in 17 - x - 3x^2$$

$$17 \hskip0.5in -x^{100}$$

$$\dfrac{1}{2}x^7 + 4x^3 - 42 \hskip0.5in \pi x^8 - \sqrt{2}x$$

Notice that 17 --- a number by itself --- is considered a polynomial.

The following are not polynomials:

$$\dfrac{1}{x + 2} \hskip0.5in 2 - \sqrt{x}$$

$$54 \sin x \hskip0.5in \dfrac{x + 15}{x - 32}$$

$$17 x^{-5} \hskip0.5in x^7 + \dfrac{2}{x^7} \quad\halmos$$


You can also have polynomials in several variables. Just as with polynomials in one variable, you can have a sum of terms, each consisting of a product of positive integer powers of the variables and numbers.


Example. Here are some polynomials:

$$xy + x^2 - 15y^4 \hskip0.5in u^2v^2 - u + 56v^4 + 13$$

The following are not polynomials:

$$\dfrac{v}{u} \hskip0.5in 2 - \sqrt{u^2 + v^2} \quad\halmos$$


For the most part, I'll concentrate on polynomials in one variable.

You add and subtract polynomials by grouping like terms, then adding or subtracting.


Example.

$$(x^2 - 2x + 3) + (5x^2 + 13) = 6x^2 - 2x + 16$$

$$(24x^3 - x + 1) - (32x^2 - 7x + 15) = 24x^3 - 32x^2 + 6x - 14$$

$$(x^2y + 3xy - 7xy^2) - (5x^2y + 6xy^2 - x^2y^2 + 4) = -4x^2y + 3xy - 13xy^2 + x^2y^2 - 4\quad\halmos$$


If $p(x)$ is a polynomial in the variable x, the value of $p(x)$ at $x = c$ is found by plugging c in for x in $p(x)$ . The value of $p(x)$ at $x = c$ is denoted $p(c)$ .


Example. Find the value of $p(x) = x^3 - 2x^2 - 7x + 14$ when $x =
   -1$ and when $x = 2$ .

$$p(-1) = (-1)^3 - 2(-1)^2 - 7(-1) + 14 = -1 - 2 + 7 + 14 = 18.$$

$$p(2) = 2^3 - 2(2^2) - 7(2) + 14 = 8 - 8 - 14 + 14 = 0.$$

If $p(c) = 0$ , then c is a root of $p(x)$ . In this case, 2 is a root of $p(x) = x^3 - 2x^2 - 7x + 14$ .


You can multiply polynomials vertically, the way you multiply numbers.


Example.

$$\matrix{& & x & + & 3 \cr & & x & - & 4 \cr \noalign{\vskip2pt\hrule\vskip2pt} & & -4x & - & 12 \cr x^2 & + & 3x & & \cr \noalign{\vskip2pt\hrule\vskip2pt} x^2 & - & x & - & 12 \cr}$$

Thus,

$$(x + 3)(x - 4) = x^2 - x - 12.\quad\halmos$$


Example.

$$\matrix{& & 2x & + & 3 \cr & & -x & + & 2 \cr \noalign{\vskip2pt\hrule\vskip2pt}\cr & & 4x & + & 6 \cr -2x^2 & - & 3x & &\cr \noalign{\vskip2pt\hrule\vskip2pt}\cr -2x^2 & + & x & + & 6 \cr}$$

Thus,

$$(2x + 3)(-x + 2) = -2x^2 + x + 6.\quad\halmos$$


Example.

$$\matrix{& & x & + & 2 \cr & & 2x & - & 5 \cr \noalign{\vskip2pt\hrule\vskip2pt}\cr & & -5x & - & 10 \cr 2x^2 & + & 4x & &\cr \noalign{\vskip2pt\hrule\vskip2pt}\cr 2x^2 & - & x & - & 10 \cr}$$

Hence,

$$(x + 2)(2x - 5) = 2x^2 - x - 10.\quad\halmos$$


Example.

$$\matrix{& & x^2 & + & x & + & 1 \cr & & & & x & - & 1 \cr \noalign{\vskip2pt\hrule\vskip2pt}\cr & & -x^2 & - & x & - & 1 \cr x^3 & + & x^2 & + & x & & \cr \noalign{\vskip2pt\hrule\vskip2pt}\cr x^3 & & & & & - & 1 \cr}$$

Therefore,

$$(x - 1)(x^2 + x + 1) = x^3 - 1.$$

The last example is a special case of a factoring formula I'll discuss later.


Example. You can also multiply polynomials involving more than one variable.

$$\matrix{& & 2x & + & 3y \cr & & x & - & 6y \cr \noalign{\vskip2pt\hrule\vskip2pt}\cr & & -12xy & - & 18y^2 \cr 2x^2 & + & 3xy & &\cr \noalign{\vskip2pt\hrule\vskip2pt}\cr 2x^2 & - & 9xy & - & 18y^2 \cr}$$

Thus,

$$(2x + 3y)(x - 6y) = 2x^2 - 9xy - 18y^2.\quad\halmos$$


You can also multiply polynomials horizontally. Consider the case $(ax + b)(cx + d)$ . The FOIL method produces the product term-by-term:

$$\hbox{\epsfysize=3in \epsffile{polynomials1.eps}}$$

Thus,

$$(ax + b)(cx + d) = ac\,x^2 + ad\,x + bc\,x + bd.$$


Example.

$$(x - 2)(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10.\quad\halmos$$


Example.

$$(2x - 1)(3x + 7) = 6x^2 + 14x - 3x - 7 = 6x^2 + 11x - 7.\quad\halmos$$


FOIL also works with polynomials of two or more variables.


Example.

$$(x - 2y)(x + 3y) = x^2 + 3xy - 2xy - 6y^2 = x^2 + xy - 6y^2. \quad\halmos$$


Some special forms come up often enough that it's worth listing them.

1. $(a - b)(a + b) = a^2 - b^2$

2. $(a - b)^2 = a^2 - 2ab + b^2$

3. $(a + b)^2 = a^2 + 2ab + b^2$ .


Example.

$$(x + 1)(x - 1) = x^2 - 1.\quad\halmos$$


Example.

$$(x + 2)^2 = x^2 + 4x + 4.\quad\halmos$$


Example.

$$(x - \sqrt{2})^2 = x^2 - 2\sqrt{2}x + 2.\quad\halmos$$


Example.

$$(2y - 5)^2 = 4y^2 - 20y + 25.\quad\halmos$$


Example.

$$(x^2 + y^2)(x^2 - y^2) = x^4 - y^4.\quad\halmos$$


Example.

$$(x - 5y)^2 = x^2 - 10xy + 25y^2.\quad\halmos$$


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