8/14/2019 Polynomials, Linear Equations

http://slidepdf.com/reader/full/polynomials-linear-equations 1/5

PolynomialsPolynomials are sums of the "variables and exponents" expressions. Each piece of the polynomial, eachpart that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, nofractional powers, and no variables in the denominator of any fractions. Here are some examples:

6 x –2 This is NOTa polynomial term...

...because the variable hasa negative exponent.

 1/ x2This is NOT

a polynomial term......because the variable is

in the denominator.

  sqrt ( x)This is NOT

a polynomial term......because the variable is

inside a radical.

4 x2 This IS a polynomial term......because it obeys all the

rules.

Here is a typical polynomial:

Notice the exponents on the terms. The first term has an exponent of 2; the second term has an

"understood" exponent of 1; and the last term doesn't have any variable at all. Polynomials are usuallywritten this way, with the terms written in "decreasing" order; that is, with the largest exponent first, thenext highest next, and so forth, until you get down to the plain old number.

Any term that doesn't have a variable in it is called a "constant" term because, no matter what value you

may put in for the variable  x, that constant term will never change. In the picture above, no matter what

 x might be, 7 will always be just 7.

The first term in the polynomial, when it is written in decreasing order, is also the term with the biggestexponent, and is called the "leading term".

The exponent on a term tells you the "degree" of the term. For instance, the leading term in the abovepolynomial is a "second-degree term" or "a term of degree two". The second term is a "first degree" term.

The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a"second-degree polynomial". Here are a couple more examples:

When a term contains both a number and a variable part, the number part is called the "coefficient". Thecoefficient on the leading term is called the "leading" coefficient.

8/14/2019 Polynomials, Linear Equations

http://slidepdf.com/reader/full/polynomials-linear-equations 2/5

In the above example, the coefficient of the leading term is 4; the coefficient of the second term is 3; the

constant term doesn't have a coefficient.  Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

The "poly" in "polynomial" means "many". I suppose, technically, the term "polynomial" should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillionterms. However, the shorter polynomials do have their own names:

• a one-term polynomial, such as 2 x or 4 x2, may also be called a "monomial" ("mono" meaning"one")

• a two-term polynomial, such as 2 x +  y or  x2 – 4, may also be called a "binomial" ("bi" meaning

"two")

• a three-term polynomial, such as 2 x +  y +  z  or  x4 + 4 x2 – 4, may also be called a "trinomial" ("tri"meaning "three")

LINEAR EQUATIONS IN TWO

VARIABLES

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

Linear equations can have one or more variables. Linear equations occur with great

regularity in applied mathematics. While they arise quite naturally when modeling many

phenomena, they are particularly useful since many non-linear equations may be

reduced to linear equations by assuming that quantities of interest vary to only a small

extent from some "background" state.

A common form of a linear equation in the two variables  x and y is

where m and b designate constants. The origin of the name "linear" comes from the

fact that the set of solutions of such an equation forms a straight line in the plane.

In this particular equation, the constant m determines the slope or gradient of that

8/14/2019 Polynomials, Linear Equations

http://slidepdf.com/reader/full/polynomials-linear-equations 3/5

line, and the constant term b determines the point at which the line crosses the y -

axis.

Since terms of a linear equations cannot contain products of distinct or equal

variables, nor any power (other than 1) or other function of a variable, equations

involving terms such as xy , x 2, y 1/3, and sin( x ) are nonlinear .

General form

Where A and B are not both equal to zero. The equation is usually written so

that A ≥ 0, by convention. The graph of the equation is a straight line, and every

straight line can be represented by an equation in the above form. If  A is

nonzero, then the x -intercept, that is the x -coordinateof the point where the graph

crosses the x -axis (y is zero), is −C / A. If B is nonzero, then the y -intercept, that isthe y -coordinate of the point where the graph crosses the y -axis (x is zero), is

−C /B, and the slope of the line is − A/B.

Standard form

where A, B, and C are integers whose greatest common factor is 1, A and B are

not both equal to zero, and A is non-negative (and if  A = 0 then B has to be

positive). The standard form can be converted to the general form, but not

always to all the other forms if  A or B is zero. It is worth noting that, while the

term occurs frequently in school-level US textbooks, it makes little mathematical

sense since most lines cannot be described by such equations. For instance, the

line cannot be described by a linear equation with integer 

coefficients since is irrational.

SYSTEMS OF EQUATIONS in TWO VARIABLES

A system of equations is a collection of two or more equations

with the same set of unknowns. In solving a system of equations,we try to find values for each of the unknowns that will satisfyevery equation in the system.

 The equations in the system can be linear or non-linear. Thistutorial reviews systems of linear equations.

8/14/2019 Polynomials, Linear Equations

http://slidepdf.com/reader/full/polynomials-linear-equations 4/5

A problem can be expressed in narrative form or the problem canbe expressed in algebraic form.

Suppose there are two linear equations with two variables x and ysuch as:

ax+by=r

cx+dy=s

where a,b,c and d are coefficients of x and y

then we can solve these two equations for the value of x and y by

the following methods:-

1)Substitution Method:-

In this method, first we find the value of x in terms of y from any

of the given equations and then substitute this value in other

eqaution, then the other equation becomes as linear equation in

one variable y and we can easily find the value of y and then find

x by substituting the value of y in any of given equation.

2)Equating the Coefficient Method:-

In this method, we equate the coefficient of x in both equations

by multiplying one equation by p and other with q such that

apx=cqx.

after it,we subtract any one eqation from other and the equation

again converts into linear equation in one variable y and we can

easily find the value of y and then find x by substituting the value

of y in any of given equation.

For your practice here is an example of two linear equations in

two variables,

 just try to solve them for the values of x and y:

8/14/2019 Polynomials, Linear Equations

http://slidepdf.com/reader/full/polynomials-linear-equations 5/5

2x+5y=24

7x+3y=26

Answer:- x=2 and y=4