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 - 4x^{2} + 2x - 4 + 3x^{3} + x^{2} - 4x - 5x^{3} + 5x^{2}

Solution: Grouping Like terms at one place, we get The given Expression = (3x^{3} - 5x^{3}) + (x^{2} - 4x^{2} + 5x^{2}) + (2x - 5x) +(5 - 4) = -2x^{3} + 2x^{2} + 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 -2x^{3} + 2x^{2} + 3x + 1which is obviously in descending order. (ii) The Expression in ascending order is 1 + 3x + 2x^{2} -2x^{3}.

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 + ax^{2} where c, b and a are real number coefficients. (iii) The general form of nth degree Polynomial in ascending order is a_{0} + a_{1}x + a_{2}x^{2} + .....................+ a_{n-2}x^{n-2} + a_{n-1}x^{n-1} + a_{n}x^{n} where a_{0}, a_{1}, a_{2},.......a_{n-2}, a_{n-1}, a_{n} are real number coefficients.
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