Polynomials

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Polynomials:

Polynomials

Index:

Index 1 . Polynomial 2. Linear Equation on Two Variables A . Its Contents A. System of Equations B . A polynomial can have B. Cartesian Plane C . Constants C. Graphing Pairs of Equation on Cartesian Plane D . Variables D. Finding Intercepts E . Exponents E. Using Table to List Solution F . Coefficients F. Special Lines G . Degree of Polynomials G. Slope H . Types of Polynomials (On Degree) H. Graphical Solution Of Linear Equations I . Linear Polynomials I. Substitution Method J . Quadratic Polynomials J. Elimination Method K . Cubic Polynomials K. Cross Multiplication Method L . Biquadratic Polynomials M . Types of Polynomials (On Term) N . Zero Polynomials O . Monomials, Binomials & Trinomials

Polynomial:

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. - a n x n +a 2 x 2 +a 1 x+a 0

Polynomial Can Have:

Polynomial Can Have A polynomial can have: Constants Variables Exponents Coefficients

Constants:

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. 2x 2 +11y-22=0 , here -22 is a constant.

Variables:

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. 10x 2 +5y=2 , here x and y are variable.

Exponents:

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.-

Coefficients:

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.- 7 x 2 − 3 xy + 15 + y Here 7, -3, 1 are the coefficients of x 2 , xy and y respectively.

Degree of Polynomials :

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 − 5 x + 2 x 5 − 7 x 9 has degree 9 .

Types Of Polynomial:

Types Of Polynomial Polynomials classified by degree – Degree Name Example −∞ Zero 0 0 (Non-zero) Constant 1 1 Linear X+1 2 Quadratic X 2 +1 3 Cubic X 3 +1 4 Quartic(Biquadratic) X 4 +1 5 Quintic X 5 +1 6 Sextic X 6 +1 7 Septic X 7 +1 8 Octic X 8 +1 9 Nonic X 9 +1 10 Decic X 10 +1 100 Hectic X 100 +1

Linear Polynomials:

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

Quadratic Polynomials:

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.- x 2 − 4 x + 7 is a quadratic polynomial, while x 3 − 4 x + 7 is not.

Cubic Polynomials:

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.- x 3 + 11 x = 9 x 2 + 55 x 3 + x 2 +10x = 20

Biquadratic Polynomials :

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.- 4x 4 + 5x 3 – x 2 + x - 1 9y 4 + 56x 3 – 6x 2 + 9x + 2

Types Of Polynomial:

Types Of Polynomial Polynomial can be classified by number of non-zero term Number of non- zero terms Name Example 0 Zero Polynomial 0 1 Monomial X 2 2 Binomial X 2 +1 3 Trinomial X 3 +1

Zero Polynomials:

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 ∞ .

Monomial, Binomial & Trinomial:

Monomial, Binomial & Trinomial Monomial :- A polynomial with one term . E.g. - 5 x 3 , 8 , and 4 xy . Binomial :- A polynomial with two terms which are not like terms. E.g. - 2 x – 3, 3 x 5 +8 x 4 , and 2 ab – 6 a 2 b 5 . Trinomial :- A polynomial with three terms which are not like terms. E.g. - x 2 + 2x - 3, 3x 5 - 8x 4 + x 3 , and a 2 b + 13x + c .

LINEAR EQUATION ON TWO VARIABLES:

L INEAR E QUATION O N T WO V ARIABLES

Slide 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

Slide 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 .

Cartesian Plane:

Cartesian Plane x- axis y-axis Quadrant I (+,+) Quadrant II ( - ,+) Quadrant IV (+, - ) Quadrant III ( - , - ) origin

Graphing Ordered Pairs on a Cartesian Plane:

Graphing Ordered Pairs on a Cartesian Plane x- axis y-axis Begin at the origin. Use the x-coordinate to move right (+) or left (-) on the x-axis. From that position move either up(+) or down(-) according to the y-coordinate . Place a dot to indicate a point on the plane. Examples: (0 , - 4) ( 6, 0) (- 3 , - 6) (6,0) (0,-4) (-3, -6)

Graphing More Ordered Pairs from our Table for the equation :

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 .

Graphing Linear Equations in Two Variables:

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

Graphing Linear Equations in Two Variables:

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

Finding INTERCEPTS:

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).

Using Tables to List Solutions:

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), … } x y 0 -4 6 0 3 -2 3/2 -3 -3 -6 -6 -8 … …

Special Lines:

Special Lines y = # is a horizontal line x = # is a vertical line y x y x

Given 2 collinear points, find the slope.:

Given 2 collinear points, find the slope. Find the slope of the line containing (3,2) and (-1,5). Slope

GRAPHICAL SOLUTIONS OF A LINEAR EQUATION:

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.

Slide 31:

For the equation 2x –y = -1 ---(1) 2x +1 = y Y = 2x + 1 X 0 2 Y 1 5 X 3 -1 Y 0 6

Slide 32:

X X ´ Y Y ´ (2,5) (-1,6) (0,3) (0,1) X = 1 Y=3

ALGEBRAIC METHODS OF SOLVING SIMULTANEOUS LINEAR EQUATIONS:

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

ELIMINATION BY SUBSTITUTION:

ELIMINATION BY SUBSTITUTION STEPS: Obtain the two equations. Let the equations be a 1 x + b 1 y + c 1 = 0 ----------- (I ) a 2 x + b 2 y + c 2 = 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

SUBSTITUTION METHOD:

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)

SUBSTITUTION METHOD:

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 ,

SUBSTITUTION METHOD:

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 )

ELIMINATION METHOD:

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

Slide 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

Slide 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

Cross Multiplication Method :

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:

Cross Multiplication Method:

Cross Multiplication Method Now, Then, ad = bc Let us now take some examples, 2x+3y=46 & 3x+5y=74

Cross Multiplication Method:

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, a 1 =2, b 1 =3, c 1 =-46 & a 2 =3, b 2 =5, c 2 =-74

Cross Multiplication Method:

Cross Multiplication Method 1 st 2 nd 3 rd

Cross Multiplication Method:

Cross Multiplication Method Taking eq. 1 st and eq. 3 rd together, we get

Cross Multiplication Method:

Cross Multiplication Method Taking eq. 2 nd and eq. 3 rd together, we get So, value of x and y by cross multiplication method is 8 and 10 respectively.

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