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

Before proceeding furher, please study Polynomials in Basic Algebra, in which we cover the basics under the heading Algebraic Expressions, if you have not already done so. The knowledge of various terms discussed there such as Variables and Constants, Coefficient, Like and Unlike terms, Multinomial, Value of an Algebraic Expression, Equation, Solution or Root of an Equation etc is a prerequisite here and in Algebra Factoring. The Formula you learnt in Arithmetic, Dividend = Divisor x Quotient + Remainder is applied here in Algebra.

Definition :

An algebraic expression in which the exponents of the variable(s) involved are whole numbers (zero or positive integers), is called a Polynomial.

General form :

An algebraic expression of the form
a + bx + cx² + dx³ + ............ is called a polynomial in single variable x, briefly a Polynomial. Here, a, b, c, d, .........are constants which are real numbers.
If all the coefficients a, b, c, d, ......... are zero, the expression becomes zero and is called zero polynomial.
If all the coefficients b, c, d, ......... other than the constant term a are zero, the expression becomes a constant (= a) and is called constant polynomial.

Please note that, for an Algebraic Expression to be called Polynomial, the exponents of the variable(s) have to be whole numbers (zero or positive integers) only. If the exponents of the variable(s) are negative integers or fractions or any real numbers other than whole numbers, then it is NOT Polynomial and will be called multinomial.

All Polynomials are multinomials, but all multinomials are not Polynomials.

Degree :

The highest exponent of the variable in a polynomial is called its Degree.

Example 1 :

Find the degree of the Expressionss
(i) 1 + 2x + 3x² (ii) y + y³ (iii) 3 + 6z² + 9z⁴

Solution:
(i) Degree of 1 + 2x + 3x² is 2.
(ii) Degree of y + y³ is 3.
(iii) Degree of 3 + 6z² + 9z⁴ is 4.

Example 2 :

Write the general form of Polynomial of
(i) first degree (ii) second degree (iii) nth degree.

Solution:
(i) The general form of first degree Polynomial is ax + b
where a and b are real number coefficients.
(ii) The general form of second degree Polynomial is ax² + bx + c where a, b and c are real number coefficients.
This is also called quadratic expression.
(iii) The general form of nth degree Polynomial is
a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + .....................+a₂x² + a₁x + a₀
where a_n, a_n-1, a_n-2,.......a₂, a₁, a₀ are real number coefficients.

Degree when there are two or more variables:

The degree of any term of a polynomial with two or more varibles
is the sum of the exponents of all the variables in that term.

Example 3 :

Write the degree of each term in the expression
2x + 3y² + 4xy + 3x²y³,

Solution:
degree of the term 2x = exponent of x = 1
degree of the term 3y² = exponent of y = 2
degree of the term 4xy = exponent of x + exponent of y = 1 + 1 = 2
degree of the term 3x²y³ = exponent of x + exponent of y = 2 + 3 = 5.

The Degree of a polynomial with two or more varibles is the degree of the highest degree term in it.

Example 4 :

Write the degree of the expression in Example 3.

Solution:
In the above example (i.e. in the expression 2x + 3y² + 4xy + 3x²y³), degree of the expression = degree of the highest degree term in it = 5.

Zero(s) :

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

Thus finding Zero of first degree Polynomial is nothing but solving the Linear Equation formed by equating it to zero. For solving, you may see
Linear Equations. Similarly, finding Zeros of second degree Polynomial (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 as Similar 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 5 :

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 right are arranged in the descending (ascending) order, then the Expression is said to be in descending (ascending) order.

Example 6 :

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 + 1 which is obviously in descending order.
(ii) The Expression in ascending order is 1 + 3x + 2x² -2x³.

Example 7 :

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 + .....................+ a_n-2x^n-2 + a_n-1x^n-1 + a_nxⁿ
where a₀, a₁, a₂,.......a_n-2, a_n-1, a_n are real number coefficients.

Addition :

While adding, similar terms are grouped together and each set of similar terms is simplified to a single term.

Example 8 :

Add 5x² + 3x - 7 and 2x² - 4x + 9

Solution:
The sum of the given Expressions = (5x² + 3x - 7) + (2x² - 4x + 9)
Grouping similar terms, we get
sum = (5x² + 2x²) + (3x - 4x ) + (-7 + 9)
= (5 + 2)x² + (3 - 4)x + (2)
= 7x² - x + 2 Ans.
The above method of adding is called horizontal method.

We can also follow another method called column method in which addition can be done quickly, accurately and easily.

Step 1: First write the Expressions in descending order, if they are not already there in that order. Here the given polynomials are already in descending order.

Step 2: Now write down one expression below the other in order and then add.
5x² + 3x - 7
2x² - 4x + 9
------------------
7x² - x + 2 Ans.
------------------

Example 9 :

Add the following Expressions by column method.
5x³ - 3x + 4 and -9x³ + 12x² -3

Solution:
Step 1: First write the Expressions in descending order. If any term is missing write the coefficient 0 for it. Here first Expression does not have x² term. So take it as 0.x². Similarly second Polynomial does not have x term. So take it as 0.x.

Step 2: Now write down one Expression below the other in order and then add.
5x³ + 0.x² - 3x + 4
-9x³ + 12x² + 0.x - 3
-----------------------------
-4x³ + 12x² - 3x + 1. Ans.
-----------------------------

Subtraction :

Subtraction is similar to addition, except that the signs of the terms of the Expression to be subtracted are to be changed.

Example 10 :

Take the two Expressions in Example 8 and subtract the second from the first by (i) horizontal method and (ii) column method.

Solution:
Let The two Expressions be denoted by A and B.
Then A = 5x² + 3x - 7 and B = 2x² - 4x + 9.
A - B = (5x² + 3x - 7) - (2x² - 4x + 9)
= (5x² + 3x - 7) - 2x² + 4x - 9
= (5x² - 2x²) + (3x + 4x) + (-7 -9)
= 3x² + 7x -16. Ans.
In column method, we follow the same procedure as in addition above except that the signs of the terms of the expression to be subtracted (written at the bottom) are changed.
  5x² + 3x - 7
  2x² - 4x + 9
(-)    (+)    (-)
-----------------
3x² + 7x - 16 Ans.
-----------------

Example 11 :

Take the two Expressions in Example 9 and subtract
the second from the first by column method.

Solution:
We have to write the second Polynomial below the first
after arranging in descending order and puttiing 0 as
coefficient for the missing terms as done in case of addition.

Then we have to change the signs of the terms
of the bottom Polynomial and then add.

   5x³ + 0.x² - 3x + 4
  -9x³ + 12x² + 0.x - 3
(+)     (-)      (-)     (+)
---------------------------------
14x³ - 12x² - 3x + 7. Ans.
----------------------------------

Multiplication :

Whille multiplying two Polynomials, we use distributive laws.

Example 12 :

Multiply the Polynomials in Example 8.

Solution:
Let The two Polynomials be denoted by A and B.
Then A = 5x² + 3x - 7 and B = 2x² - 4x + 9.
AB = (5x² + 3x - 7)(2x² - 4x + 9)
= 5x²(2x² - 4x + 9) + 3x(2x² - 4x + 9) - 7(2x² - 4x + 9)
= 5x²(2x²) - 5x²(4x) + 5x²(9) + 3x(2x²) + 3x(-4x)
+ 3x(9) -7(2x²) -7(-4x) -7(9)
= 10x⁴ - 20x³ + 45x² + 6x³ - 12x² + 27x - 14x² + 28x - 63
= 10x⁴ + x³(-20 + 6) + x²(45 -12 -14) + x(27 + 28) - 63
= 10x⁴ - 14 x³ + 19x² + 55x + 63 Ans.

The above method is horizontal method.

We can adopt column method here also.

In this method, we write multiplicand and
the multiplier in descending powers of x,
arrange one under another,
and multiply the multiplicand
by every term of the multiplier and add.

5x² + 3x - 7
2x² - 4x + 9
--------------------------------------------------------------
10x⁴ + 6x³ - 14x²
                 - 20x³ - 12x² + 28x
                               + 45x² + 27x - 63
-----------------------------------------------------------
                      10x⁴ - 14x³ + 19x² + 55x - 63 Ans.

The degree of the product = 4 = the degree of the multiplicand (= 2) + the degree of the multiplier (= 2)
We can observe that in multiplication of Polynomials, the degree of the product equals the sum of the degrees of the multiplicand and multiplier.

Division :

Let us see the steps in the procedure of the division of Polynomials by means of an example.

Example 13 :

Divide 10x² - x + 21 by 2x - 3

Solution:

dividend
          divisor 2x - 3) 10x² -     x + 21 ( 5x + 7 quotient
10x² - 15x
-----------------------------------
             +14x + 21
             +14x - 21
-------------------------------------
                                + 42 remainder
--------------------------------------

step 1: Arrange the Polynomials in descending powers of the variable x.
step 2: Divide the first term of the dividend by the first term of the divisor i.e. 10x² ÷ 2x = 5x. Write the result as the term of the quotient.
step 3: Multiply the term of the quotient i.e. 5x with each term of the divisor i.e. 2x - 3 giving 5x(2x - 3) = 10x² - 15x and put the result under dividend for subtracting. The subtraction gives 14x.
step 4: Bring down one of the remaining terms of the dividend. This gives the remainder as 14x + 21.
step 5: Now, use the remainder 14x + 21 as new dividend and repeat the steps 2 to 4.
step 6: Stop when the remainder is zero or the degree of the remainder is less than the degree of the divisor.
∴ quotient = 5x + 7; remainder = 42.
step 7: verification Check whether dividend = divisor x quotient + remainder is satisfied or not.
Here divisor x quotient + remainder = (2x - 3)(5x + 7) + 42
= 10x² + 14x - 15x - 21 + 42
= 10x² - x +21 = dividend (verified.)

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.