loan

subject: Polynomial Identity Testing [print this page]

In mathematics, a polynomial is an expression of the finite length constructed from variables (also known as in determinate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. Polynomials identity appears in a variety of areas of mathematics and science. For example, they are used to form polynomial identity, which encode a range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions.

Various Polynomial Identity Testing:

X2-y2 = (x-y) (x + y)identity (1)

X3-y3 = (x-y)(x2+xy+y2).identity (2)

X3+ y3 = (x + y)(x2- xy + y 2)..identity (3)

(X + y)2 = x2 + 2xy + y2identity (4)

(X-y)2 = x2 - 2xy + y2 identity(5)

Examples for Various Polynomial Identity Testing:

Example testing 1:

Expand the following:

(i) (3x + 7y)2 (ii) (11a 7b)2 (iii) (2p + 5q)(2p 5q)

Solution:

(i) (3x + 7y)2 = (3x)2 + 2(3x)(7y) + (7y)2 =9x2+ 42xy + 49y2.[ using polynomial identity]

(ii) (11a 7b)2 = (11a)2 2(11a) (7b) + (7b)2 = 121a2 154ab + 49b2. [ using polynomial identity]

(iii) (2p + 5q)(2p 5q) = (2p)2 (5q)2 = 4p2 25q2 [ using polynomial identity]

Example testing 2:

Expand the following: (i) (3x+2y)3 (ii) (2x23y)3

Solution:

(i) (3x+2y)3 = (3x)3 + 3(3x)2(2y) + 3(3x)(2y)2 + (2y)3 [ using polynomial identity]

= 27x3 + 3(9x2)(2y) + 3(3x)(4y2) + 8y3

= 27x3 + 54x2y + 36xy2 + 8y3.

(ii) (2x2 3y)3 = (2x2)3 3(2x2)2 (3y) + 3(2x2) (3y)2 (3y)3 [ using polynomial identity]

= 8x6 3(4x4)(3y) + 3(2x2)(9y2) 27y3

= 8x6 36x4y + 54x2y2 27y3

Example testing 3:

If (x + p)(x + q) = x2 5x 400, find the value p2 + q2.

Solution:

By product formula, we have (x + p) (x + q) = x2 + (p + q)x + pq.[ using polynomial identity]

So, by comparison, we get p + q = 5, pq = 400.

Now, we have p2 + q2 = (p + q)2 2 pq = (5)2 2(400) = 25 + 800 = 825.

Example testing 4: Simplify x y /x3-y3

Solution:

x y/x3-y3 = x-y / (x2-xy+y2)(x-y) [ using polynomial identity]

=1 / (x2-xy+y2)

In mathematics, polynomial is one of the main divisions in maths. Here we are going to explain about how to expand polynomial expressions. Now we are going to known about expanded polynomial equations.

Standard form of polynomial is f(x) = ax^4 + bx^3 + cx^2 +dx + e . by change the values of coefficient of a, b, c, d and e.

Expanded polynomials: (x + y) ^ nth order of degree.

Example: (x + y) ^2 = x^2 + 2 x y + y ^2.

Here, we are going to see about, how to expand polynomial expression using by polynomial calculator.

Polynomial Calculator:

Here we are going to operate the polynomial calculator. In expanded polynomial, we use only positive integer, will not accept negative integer value because we will not expand polynomial equations using by order of degree is negative integer value.

Polynomial calculator:

expand

Enter the polynomial expression = e.g. (x + y)

x = Enter the order of degree (positive integer value).

Check this Binomial Expansion Formula awesome i recently used to see.

Evaluated polynomial expression value will be displayed.

Some Problems for Understanding about Polynomial Calculator:

Problem 1:

To expand the polynomial expression (x +y), using the polynomial calculator.

Solution:

Using by calculator,

Given polynomial expression is = (x +y)

x = 1

Evaluated polynomial expression = x + y

Problem 2:

To expand the polynomial expression (x + y), using the polynomial calculator.

Solution:

Using by calculator,

Given polynomial expression = (x + y)

x = 2

Evaluated polynomial expression = x^2 + 2 x y + y^2.

Problem 3:

To expand the polynomial expression (2x + 2y), using the polynomial calculator.

Solution:

Using by calculator,

Given polynomial expression = (2x +2y)

x = 2

Evaluated polynomial expression = 4x^2 + 8 x y + 4 y^2.

Problem 4:

To expand the polynomial expression (x^2 + y^2), using the polynomial calculator.

Solution:

Using by calculator,

Given polynomial expression = (x^2 +y^2)

x = 2

Evaluated polynomial expression = x^4 + 2 x^2 y^2 + y^4.

Problem 5:

To expand the polynomial expression (x + y +1), using the polynomial calculator.

Solution:

Using by calculator,

Given polynomial expression = (x + y +1)

x = 2

Evaluated polynomial expression = 1+ 2x + x^2 + 2y +2 x y + y^2.

Practices problems for understanding the polynomial calculator:

Problem 1:

If x =1, to expand the polynomial expression (x - y -1)

Answer = x-y-1.

Problem 2:

If x = 2, to expand the polynomial expression (x^2 y^2)

Answer = x^4-2 x^2 y^2 + y^4.

Problem 3:

If x = 3, to expand the polynomial expression (2x+2y)

Answer = 8x^3+24 x^2 y +24 x y^2 +8y^3.

by: mathqa

welcome to loan (http://www.yloan.com/)