POLYNOMIALS - DEFINITION, DEGREE, ZEROS, BASIC OPERATIONS, EXAMPLES, LINKS

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Before proceeding furher, please study

Definition, General form, Degree of Polynomials,
if you have not already done so.

That knowledge is prerequisite here.

Here, we cover Zeros (Linear Equations,
Quadratic Equations), Simplified form
(grouping all like terms), Arranging
in the descending or ascending powers
of the variable and provide Links for
Adding and Subtracting,Multiplying
and Dividing, Factoring etc.





Zero(s) :

The value(s) of the variable for which the value of the Polynomial is zerois(are) called Zero(s) of the Polynomial.

From this it is clear that finding Zero(s) is nothing butsolving the equation formed by equating it to zero.

Thus finding Zero of first degree Expresion is nothing butsolving the Linear Equation formed by equating it to zero. For solving, you may see
Linear Equations. Similarly, finding Zeros of second degree Expression (Quadratic Expression) is nothing but solving the Quadratic Equation formed by equating the second degree expression(Quadratic Expression) to zero. For solving, you may see
Quadratic Equations.











Simplified form :

Please Refer to the Like terms and Unlike terms discussed with examples under the heading Algebraic Expressions in Basic Algebra,
if you have not already done so.
Terms differing only in coefficients are like terms. They are also known asSimilar terms. Unlike terms are also called dissimilar terms.

If no two terms of a Polynomial are alike, then it is said to be in the simplified form or standard form

Example 1 :

Write the following Expression in the simplified (standard) form.
5 - 4x2 + 2x - 4 + 3x3 + x2 - 4x - 5x3 + 5x2

Solution:
Grouping Like terms at one place, we get
The given Expression
= (3x3 - 5x3) + (x2 - 4x2 + 5x2) + (2x - 5x) +(5 - 4)
= -2x3 + 2x2 + 3x + 1. Ans.









Arranging in the descending or ascending powers of the variable:

If the degrees of the different terms in the Expression from left to rightare arranged in the descending (ascending) order, then the Expression is said to bein descending (ascending) order.

Example 2 :

Arrange the Answer for the problem in Example 5 (i) in descending order.(ii) in ascending order.

Solution:
(i) The answer given in the previous example is -2x3 + 2x2 + 3x + 1which is obviously in descending order.
(ii) The Expression in ascending order is 1 + 3x + 2x2 -2x3.

Example 3 :

Consider the question of Example 2 with slight modification.
Write the general form of Polynomial in ascending order of
(i) first degree (ii) second degree (iii) nth degree.

Solution:
(i) The general form of first degree Polynomial in ascending order is b + ax where b and a are real number coefficients.
(ii) The general form of second degree Polynomial in ascending order is c + bx + ax2 where c, b and a are real number coefficients.
(iii) The general form of nth degree Polynomial in ascending order is
a0 + a1x + a2x2 + .....................+ an-2xn-2 + an-1xn-1 + anxn
where a0, a1, a2,.......an-2, an-1, an are real number coefficients. Great Deals on School & Homeschool Curriculum Books

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Addition, Subtraction, Multiplication, Division :

The following Links take you to the above topics.

Adding and Subtracting

Multiplying and Dividing

Great deals on School & Homeschool Curriculum Books and Software

Algebraic Identities :

Our Study here include Algebraic
Identities which are very useful in evaluating
some given expressions and in factorization.

For the study of them, go to

Algebra Formulas.









Remainder Theorem, Factor Theorem, Factoring :

Our Study here also include
Remainder theorem, Factor Theorem
and Factoring.

All these topics are covered in the following Links.

Algebra Factoring.

Factoring Special products.

Factoring Trinomials (Quadratics).

Factoring Polynomials.

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