Polynomials





Definition of polynomials:

Polynomials is the algebraic topic. An algebraic expression, which have only non negative powers of the variables is called polynomial.

In an expression, generally we have variables and constant terms.

Example  3 x² + 2 x + 6

Here x is variable and 6 without any variant is called Constant.

Degree :

The degree of a algebraic equation is the highest degree for a term with non-zero coefficient. The degree of a term is sum of the powers of each variable in the term.

Example

4x⁷-5

In this expression we have only one term and the power of the term is 7, so the expression has deg 7

5 x³ y³ - 4 x² y² + 7 x y

Here, it has 3 terms and power of the first term is 3+3=6, the power of the second term is 2+2=4, and the power of the last term is 1+1=2, here the highest power is 6, so the expression has deg 3+3=6

Type of the algebraic equation Meaning
Monomial
Expression which is having only one term is known as monomial
Example: 3x,5a,... etc
Binomial
 Expression which is having two terms is known as binomial
Example: 3x - 2, 3 x ² + 7
Trinomial
Expression which is having three terms is known as trinomial
Example: 4x²+ 7 x - 6, 5y³+ 2 y - 6

We can do basic arithmetic operations with two algebraic equation. That is we can add, subtract, multiply or divide any two polynomial.

Addition:

For adding any two polynomials we have to combine the like terms.

Add 4 x² + 7 x - 6, x² - 3 x - 2

For adding these two poly-nomials we have to combine the like terms. Here the like terms are 4x² and x², 7x and -3x , -6 and -2

If we combine 4x² and x² we will get 5x²

If we combine 7x and -3x we will get 4x

If we combine -6 and 2 we will get -4

So the final answer is 5x² + 4x - 4

Subtraction:

(2 x³ - 2 x² + 4 x - 3)- (x³ + x² - 5 x + 4)

Step 1:

In the first step we are going to multiply the negative with inner terms.

                    = 2x³ -2x² + 4x -3 - x³-x²+5x-4
Step 2:

In the second step we have to combine the like terms

                   = 2x³ - x³ -2x²-x² + 4x + 5x - 3 - 4
Step 3:

After combining the like terms we will get the answer

                  = x³ + x² + 9x - 7

Multiplication :

there are two formats for this: horizontal and vertical, like in addition.

The simplest case of multiplication of polynomials is multiplication of monomials.

For instance :

Simplify: ( 5 x² )(-2x ³)

For multiplying these two monomials  we have to just multiply the numbers and add the powers using the exponent rule.

So (-6x²)(3x³) = -18 x² ⁺ ³= -18 x⁵

Division:

Division of polynomials involves two cases, the first one is simplification,which is reducing the fraction and the second one is long division.


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