Polynomials
Index
Polynomial
A polynomial is a mathematical expression
involving a sum of powers in one or more variables
multiplied by coef...
Polynomial Can Have
A polynomial can have:
Constants
Variables
Exponents
Coefficients
Constants
In mathematics, a constant is a non-
varying value, i.e. completely fixed or fixed
in the context of use. The te...
Variables
In mathematics, a variable is a value that
may change within the scope of a given
problem or set of operations. ...
Exponents
Exponents are sometimes referred to as
powers and means the number of times the
'base' is being multiplied. In t...
Coefficients
For other uses of this word, see coefficient
(disambiguation).In mathematics, a coefficient is a
multiplicati...
Degree of Polynomials
The degree of a polynomial is the highest degree for
a term.
The degree of a term is the sum of the ...
Types Of Polynomial
Polynomials classified by degree –
Linear Polynomials
In a different usage to the above, a
polynomial of degree 1 is said to be linear,
because the graph of ...
Quadratic Polynomials
In mathematics, a quadratic polynomial or
quadratic is a polynomial of degree two, also called
secon...
Cubic Polynomials
Cubic polynomial is a polynomial of having
degree of polynomial no more than 3 or
highest degree in the ...
Biquadratic Polynomials
Biquadratic polynomial is a polynomial of
having degree of polynomial is no more
than 4 or highest...
Types Of Polynomial
Polynomial can be classified by number of non-zero term
Zero Polynomials
The constant polynomial whose coefficients are all
equal to 0. The corresponding polynomial function
is t...
Monomial, Binomial & Trinomial
Monomial:-
A polynomial with one term.
E.g. - 5x3
, 8, and 4xy.
Binomial:-
A polynomial wit...
LINEAR
EQUATION ON
TWO VARIABLES
A pair of linear equations in two variables is
said to form a system of simultaneous linear
equations.
For Example, 2x – 3...
The general form of a linear equation in two
variables x and y is
ax + by + c = 0 , a ≠ 0, b ≠ 0, where
a, b and c being r...
Cartesian Plane
x- axis
y-axis
Quadrant I
(+,+)
Quadrant II
( - ,+)
Quadrant IV
(+, - )
Quadrant III
( - , - )
origin
Graphing Ordered Pairs on a Cartesian
Plane
x- axis
y-axis
1) Begin at the origin.
2) Use the x-coordinate
to move right (...
Graphing More Ordered Pairs from our
Table for the equation
x
y
(3,-2)
(3/2,-3)
(-6, -8)
2x – 3y = 12
•Plotting more point...
Graphing Linear Equations
in Two Variables
 The graph of any linear
equation in two variables
is a straight line.
 Findi...
Graphing Linear Equations
in Two Variables
• On our previous graph,
y = 2x – 3y = 12, find the
intercepts.
• The x-interce...
Finding INTERCEPTS
To find the
x-intercept: Plug in
ZERO for y and
solve for x.
2x – 3y = 12
2x – 3(0) = 12
2x = 12
x = 6...
Using Tables to List Solutions
 For an equation
we can list some
solutions in a table.
 Or, we may list the
solutions in...
Special Lines
y = # is a horizontal line x = # is a vertical line
y
x
y
x
Given 2 collinear points, find the slope.
Find the slope of the line containing
(3,2) and (-1,5).
( )
2 1
2 1
2 5 3
3 1 4
...
GRAPHICAL SOLUTIONS OF A
LINEAR EQUATION
Let us consider the following system of two
simultaneous linear equations in two ...
For the equation
2x –y = -1 ---(1)
2x +1 = y
Y = 2x + 1
XX´
Y
Y
´
(2,5)
(-1,6)
(0,3)(0,1)
ALGEBRAIC METHODS OF
SOLVING SIMULTANEOUS LINEAR
EQUATIONS
The most commonly used algebraic methods
of solving simultaneou...
ELIMINATION BY SUBSTITUTION
STEPS:
Obtain the two equations. Let the equations be
a1x + b1y + c1 = 0 ----------- (I )
a2x ...
SUBSTITUTION METHOD
Solve the equation obtained in the
previous step to get the value of x.
Substitute the value of x and ...
SUBSTITUTION METHOD
x + 2y = -1
x = -2y -1 ------- (III)
Substituting the value of x in equation
(II), we get
2x – 3y = 12...
SUBSTITUTION METHOD
Putting the value of y in eq. (III), we get
x = - 2y -1
x = - 2 x (-2) – 1
x = 4 - 1
x = 3
Hence the s...
ELIMINATION METHOD
In this method, we eliminate one of the two
variables to obtain an equation in one variable
which can e...
Let 3x + 2y = 11 --------- (I)
2x + 3y = 4 ---------(II)
Multiply 3 in equation (I) and 2 in equation (ii) and
subtracting...
putting the value of y in equation (II) we get,
2x + 3y = 4
2 x 5 + 3y = 4
10 + 3y = 4
3y = 4 – 10
3y = - 6
y = - 2
Hence,...
Cross Multiplication Method
In elementary arithmetic, given an equation
between two fractions or rational
expressions, one...
Cross Multiplication Method
Now,
Then, ad = bc
Let us now take some examples,
2x+3y=46
& 3x+5y=74
Cross Multiplication Method
2x+3y=46, i.e., 2x+3y-46=0
3x+5y=74, i.e., 3x+5y-74=0
We know that equation for this method is...
Cross Multiplication Method
Cross Multiplication Method
Taking eq. 1st
and eq. 3rd
together, we get
Cross Multiplication Method
Taking eq. 2nd
and eq. 3rd
together, we get
So, value of x and y by cross multiplication
metho...
Polynomials And Linear Equation of Two Variables
of 47

Polynomials And Linear Equation of Two Variables

A complete description of polynomials and also various methods to solve the Linear equation of two variables by substitution, cross multiplication and elimination methods. For polynomials it also contains the description of monomials, binomials etc.
Published on: Mar 4, 2016
Published in: Education      Technology      
Source: www.slideshare.net


Transcripts - Polynomials And Linear Equation of Two Variables

  • 1. Polynomials
  • 2. Index
  • 3. Polynomial A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by e.g. - anxn +a2x2 +a1x+a0
  • 4. Polynomial Can Have A polynomial can have: Constants Variables Exponents Coefficients
  • 5. Constants In mathematics, a constant is a non- varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable (i.e. variable quantity), which is a symbol that stands for a value that may vary. For e.g. 2x2 +11y-22=0, here -22 is a constant.
  • 6. Variables In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. For e.g. 10x2 +5y=2, here x and y are variable.
  • 7. Exponents Exponents are sometimes referred to as powers and means the number of times the 'base' is being multiplied. In the study of algebra, exponents are used frequently. For e.g.-
  • 8. Coefficients For other uses of this word, see coefficient (disambiguation).In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For e.g.- 7x2 − 3xy + 15 + y Here 7, -3, 1 are the coefficients of x2 , xy and y respectively.
  • 9. Degree of Polynomials The degree of a polynomial is the highest degree for a term. The degree of a term is the sum of the powers of each variable in the term. The word degree has for some decades been favoured in standard textbooks. In some older books, the word order is used. For e.g.- The polynomial 3 − 5x + 2x5 − 7x9 has degree 9.
  • 10. Types Of Polynomial Polynomials classified by degree –
  • 11. Linear Polynomials In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line. For e.g.- 2x+1 11y +3
  • 12. Quadratic Polynomials In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. For e.g.- x2 − 4x + 7 is a quadratic polynomial, while x3 − 4x + 7 is not.
  • 13. Cubic Polynomials Cubic polynomial is a polynomial of having degree of polynomial no more than 3 or highest degree in the polynomial should be 3 and should not be more or less than 3. For e.g.- x3 + 11x = 9x2 + 55 x3 + x2 +10x = 20
  • 14. Biquadratic Polynomials Biquadratic polynomial is a polynomial of having degree of polynomial is no more than 4 or highest degree in the polynomial is not more or less than 4. For e.g.- 4x4 + 5x3 – x2 + x - 1 9y4 + 56x3 – 6x2 + 9x + 2
  • 15. Types Of Polynomial Polynomial can be classified by number of non-zero term
  • 16. Zero Polynomials The constant polynomial whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The degree of the zero polynomial is undefined, but many authors conventionally set it equal to -1 or ∞.
  • 17. Monomial, Binomial & Trinomial Monomial:- A polynomial with one term. E.g. - 5x3 , 8, and 4xy. Binomial:- A polynomial with two terms which are not like terms. E.g. - 2x – 3, 3x5 +8x4 , and 2ab – 6a2 b5 . Trinomial:- A polynomial with three terms which are not like terms. E.g. - x2 + 2x - 3, 3x5 - 8x4 + x3 , and a2 b + 13x + c.
  • 18. LINEAR EQUATION ON TWO VARIABLES
  • 19. A pair of linear equations in two variables is said to form a system of simultaneous linear equations. For Example, 2x – 3y + 4 = 0 x + 7y – 1 = 0 Form a system of two linear equations in variables x and y. System of Equations or Simultaneous Equations
  • 20. The general form of a linear equation in two variables x and y is ax + by + c = 0 , a ≠ 0, b ≠ 0, where a, b and c being real numbers. A solution of such an equation is a pair of values, one for x and the other for y, which makes two sides of the equation equal. Every linear equation in two variables has infinitely many solutions which can be represented on a certain line.
  • 21. Cartesian Plane x- axis y-axis Quadrant I (+,+) Quadrant II ( - ,+) Quadrant IV (+, - ) Quadrant III ( - , - ) origin
  • 22. Graphing Ordered Pairs on a Cartesian Plane x- axis y-axis 1) Begin at the origin. 2) Use the x-coordinate to move right (+) or left (-) on the x-axis. 3) From that position move either up(+) or down(-) according to the y- coordinate . 4) Place a dot to indicate a point on the plane. Examples: (0, -4) (6, 0) (-3, -6) (6,0) (0,-4) (-3, -6)
  • 23. Graphing More Ordered Pairs from our Table for the equation x y (3,-2) (3/2,-3) (-6, -8) 2x – 3y = 12 •Plotting more points we see a pattern. •Connecting the points a line is formed. •We indicate that the pattern continues by placing arrows on the line. •Every point on this line is a solution of its equation.
  • 24. Graphing Linear Equations in Two Variables  The graph of any linear equation in two variables is a straight line.  Finding intercepts can be helpful when graphing.  The x-intercept is the point where the line crosses the x- axis.  The y-intercept is the point where the line crosses the y- axis.  On our previous graph, y = 2x – 3y = 12, find the intercepts. y x
  • 25. Graphing Linear Equations in Two Variables • On our previous graph, y = 2x – 3y = 12, find the intercepts. • The x-intercept is (6,0). • The y-intercept is (0,-4). y x
  • 26. Finding INTERCEPTS To find the x-intercept: Plug in ZERO for y and solve for x. 2x – 3y = 12 2x – 3(0) = 12 2x = 12 x = 6 Thus, the x-intercept is (6,0). To find the y-intercept: Plug in ZERO for x and solve for y. 2(0) – 3y = 12 2(0) – 3y = 12 -3y = 12 y = -4 Thus, the y-intercept is (0,-4).
  • 27. Using Tables to List Solutions  For an equation we can list some solutions in a table.  Or, we may list the solutions in ordered pairs . {(0,-4), (6,0), (3,-2), ( 3/2, -3), (-3,-6), (-6,-8), … } 2 3 12x y− =
  • 28. Special Lines y = # is a horizontal line x = # is a vertical line y x y x
  • 29. Given 2 collinear points, find the slope. Find the slope of the line containing (3,2) and (-1,5). ( ) 2 1 2 1 2 5 3 3 1 4 y y m x x − − − = = = − − − Slop e
  • 30. GRAPHICAL SOLUTIONS OF A LINEAR EQUATION Let us consider the following system of two simultaneous linear equations in two variable. 2x – y = -1 3x + 2y = 9 Here we assign any value to one of the two variables and then determine the value of the other variable from the given equation.
  • 31. For the equation 2x –y = -1 ---(1) 2x +1 = y Y = 2x + 1
  • 32. XX´ Y Y ´ (2,5) (-1,6) (0,3)(0,1)
  • 33. ALGEBRAIC METHODS OF SOLVING SIMULTANEOUS LINEAR EQUATIONS The most commonly used algebraic methods of solving simultaneous linear equations in two variables are 1. Method of elimination by substitution 2. Method of elimination by equating the coefficient 3. Method of Cross- multiplication
  • 34. ELIMINATION BY SUBSTITUTION STEPS: Obtain the two equations. Let the equations be a1x + b1y + c1 = 0 ----------- (I ) a2x + b2y + c2 = 0 ----------- (II) Choose either of the two equations, say (I ) and find the value of one variable , say ‘y’ in terms of x Substitute the value of y, obtained in the previous step in equation (II) to get an equation in x
  • 35. SUBSTITUTION METHOD Solve the equation obtained in the previous step to get the value of x. Substitute the value of x and get the value of y. Let us take an example x + 2y = -1 ------------------ (I) 2x – 3y = 12 -----------------(II)
  • 36. SUBSTITUTION METHOD x + 2y = -1 x = -2y -1 ------- (III) Substituting the value of x in equation (II), we get 2x – 3y = 12 2 ( -2y – 1) – 3y = 12 - 4y – 2 – 3y = 12 - 7y = 14 , y = -2 ,
  • 37. SUBSTITUTION METHOD Putting the value of y in eq. (III), we get x = - 2y -1 x = - 2 x (-2) – 1 x = 4 - 1 x = 3 Hence the solution of the equation is ( 3, - 2 )
  • 38. ELIMINATION METHOD In this method, we eliminate one of the two variables to obtain an equation in one variable which can easily be solved. Putting the value of this variable in any of the given equations, the value of the other variable can be obtained. For example: we want to solve, 3x + 2y = 11 2x + 3y = 4
  • 39. Let 3x + 2y = 11 --------- (I) 2x + 3y = 4 ---------(II) Multiply 3 in equation (I) and 2 in equation (ii) and subtracting eq. iv from iii, we get
  • 40. putting the value of y in equation (II) we get, 2x + 3y = 4 2 x 5 + 3y = 4 10 + 3y = 4 3y = 4 – 10 3y = - 6 y = - 2 Hence, x = 5 and y = -2
  • 41. Cross Multiplication Method In elementary arithmetic, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. For an equation like the following:
  • 42. Cross Multiplication Method Now, Then, ad = bc Let us now take some examples, 2x+3y=46 & 3x+5y=74
  • 43. Cross Multiplication Method 2x+3y=46, i.e., 2x+3y-46=0 3x+5y=74, i.e., 3x+5y-74=0 We know that equation for this method is- So, a1=2, b1=3, c1=-46 & a2=3, b2=5, c2=-74
  • 44. Cross Multiplication Method
  • 45. Cross Multiplication Method Taking eq. 1st and eq. 3rd together, we get
  • 46. Cross Multiplication Method Taking eq. 2nd and eq. 3rd together, we get So, value of x and y by cross multiplication method is 8 and 10 respectively.