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Polynomials

April 8th, 2009 |

Author: Merrin

A polynomial is of the form

$P(x)=a_{n} x^{n} +a_{n-1} x^{n-1}+ ... +a_{1} x+a_{0}$

The numbers a_k are called the coefficients. Each piece

$a_k x^{k}$

is called a term. The number

$a_{n} \ne 0$

is called the leading coefficient which is the factor of the term which dominates the behavior when x is large.

The foil method can be used to multiply two binomials together. Foil stands for first, outer, inner, and last.

$(a+b)(c+d)$

The First terms of each binomial multiplied together are ac.

The Outer terms of each binomial multiplied together are ad.

The Inner terms of each binomial multiplied together are bc.

The Last terms of each binomial multiplied together are bd.

Add them all together and you get what you started with

$ac+ad+bc+cd=a(c+d)+b(c+d)=(a+b)(c+d)$

The foil method is nothing more than using the distributive property. Factoring

is this process in reverse.

There are some special binomials where the inner and outer terms cancel, such as the difference of squares.

$(a-b)(a+b)=a^{2} -b^{2}$

For multiplying polynomials with more terms you just use the distributive property.

$(1+x+x^{2} )(a+b+c)=(1)(a+b+c)+x(a+b+c)+x^{2} (a+b+c)$

Multiplying out products of polynomials is known as an expansion.

Sometimes it is useful to undo the expansion of a product of polynomials. This is called factoring. If the expression you expanded is large enough it won’t always be possible to factor it without a priori knowledge, but it is always possible to do the expansion. We will start with factoring binomials. If you have a binomial of the form

$x^{2} +bx+c=(x+r_{1} )(x+r_{2} )$

You can multiply out the product of the binomial and see that you need to have

$r_{1} r_{2} =c\quad r_{1} +r_{2} =b$

This is the game you need to play so if you are asked to factor

$x^{2} -4x+4$

The ways you can factor 4 to make products equal c are (1,4),(2,2),(-1,-4), and (-2,-2). We see that only in the last case do the two roots add up to b which for us in this example is -4. So the factorization is

$(x-2)(x-2)=x^{2} -4x+4$

Later we will learn the quadratic formula which allows us to factor the most general binomials.

One way to test if a quantity is a factor of a polynomial is to divide by it and see if you get no remainder. There is a also a theorem that if P(c)=0 then (x-c) is a factor of the polynomial. The most general division of two polynomials can be written as follows.

$\cfrac{P(x)}{Q(x)} =q(x)+r(x)$

The most general rational function is the ratio of two polynomials P(x)/Q(x). Division of polynomials can be used to calculate a quotient and a remainder just like for normal division of numbers. Polynomial division is set up the following way

$Q(x)\left)\vphantom{1 P(x)}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{P(x)}}$

If the numerator is of higher degree than the denominator then there is a nonzero quotient q(x). The remaining term is the remainder r(x) which is the remaining fraction after the quotient is divided out.

Example 1. Perform,

$x-3\left)\vphantom{1 x^{3} -2x^{2} +x-1}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{x^{3} -2x^{2} +x-1}}$

Solution 1.

${\cfrac{x^{3} -2x^{2} +x-1}{x-3} =x-3\mathop{\left)\vphantom{1 \begin{array}{l} {x^{3} -2x^{2} +x-1} \\ {\underline{x^{3} -3x^{2} }} \\ {\quad +x^{2} +x-1} \\ {\quad \underline{\, +x^{2} -3x}} \\ {\quad \quad \quad \quad \quad 4x-1} \\ {\quad \quad \quad \quad \quad \underline{4x-12}} \\ {\quad \quad \quad \quad \quad \quad \quad \quad 11} \end{array}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{array}{l} {x^{3} -2x^{2} +x-1} \\ {\underline{x^{3} -3x^{2} }} \\ {\quad +x^{2} +x-1} \\ {\quad \underline{\, +x^{2} -3x}} \\ {\quad \quad \quad \quad \quad 4x-1} \\ {\quad \quad \quad \quad \quad \underline{4x-12}} \\ {\quad \quad \quad \quad \quad \quad \quad \quad 11} \end{array}}}}\limits^{\displaystyle\hfill\,\, x^{2} +x+4\quad } }$

So for this rational function we can write in standard terms

$\cfrac{x^{3} -2x^{2} +x-1}{x-3} =x^{2} +x+4+\cfrac{11}{x-3}$

Polynomial division is a necessary step in partial fractions analysis to express a rational function in terms of a quotient and a remainder when the degree of the numerator is less than the degree of the denominator. Partial fractions are used in calculus so I will show you the necessary steps to evaluate them.

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Polynomials

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