Table of contents  
Introduction  
Degree of a Polynomial  
Arithmetic Operations  
Properties of Polynomial  
Zeroes of a Polynomial 
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Types of algebraic Expression
A monomial is an algebraic expression that has only one term.
Or we say a single term expression is a monomial.
The examples of monomials are: 12x, 12, 25x, 24y, 4a, xy
A binomial is an algebraic expression that has two unlike terms.
The examples of binomials are:
A trinomial is an algebraic expression that has three terms in it.
The examples of trinomials are:
The expression x + y + 3x is not a trinomial as the terms x and 3x are like terms.
There are 5 simple steps present to find the degree of a polynomial:
Example: 3x^{2}  3x^{4}  5 + 2x + 2x^{2}  x
Based on the degree of a polynomial, the polynomials in one variable are classified as follows:
If the degree of the polynomial is zero (0), then the polynomial is called zero or constant polynomial. Such kinds of polynomials have only constants. They don’t have variables.
The examples of constant polynomials are 2, 5, 7 and so on. Here, 2 can be written as 2x^{0}, 5 can be written as 5x^{0}, and so on.
If the degree of the polynomial is 1 (one), then the polynomial is called a linear polynomial. The linear polynomial in one variable has only one solution.
Examples of linear polynomials in one variable are:
A polynomial with the highest degree of 2 is called a quadratic polynomial. A quadratic polynomial in one variable has only two solutions. Some of the examples of quadratic polynomials in one variable are:
If the highest exponent of a variable in a polynomial is 3 (i.e. degree of a polynomial is 3), then the polynomial is called a cubic polynomial. A cubic polynomial in one variable has exactly 3 solutions. The examples of a cubic polynomial in one variable are:
For adding any two polynomials, we have to combine the like terms.
Example: Add 4 x² + 7 x  6, x²  3 x + 2
∵ 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
Thus, the final answer is 5x² + 4x  4.
(2 x³  2 x² + 4 x  3)  (x³ + x²  5 x + 4)
Step 1: Multiply the negative with inner terms i.e. 2x³ 2x² + 4x 3  x³x²+5x4
Step 2: Combine the like terms i.e. 2x³  x³ 2x²x² + 4x + 5x  3  4
Step 3: The answer i.e. x³  3x² + 9x  7.
There are two formats for this: horizontal and vertical, like in addition.
The simplest case of the multiplication of polynomials is the multiplication of monomials.
Example: (5x²)( 2x³)
For multiplying these two monomials, we have to just multiply the numbers and add the powers using the exponent rule.
⇒ (5x²)(2x³) = 10x² ⁺ ³= 10x⁵
The division of polynomials involves two cases, the first one is a simplification, which is reducing the fraction and the second one is a long division.
Some of the important properties of polynomials along with some important polynomial theorems are as follows:
(i) Division Algorithm
If a polynomial P(x) is divided by a polynomial G(x) results in quotient Q(x) with remainder R(x), then, P(x) = G(x) • Q(x) + R(x)
(ii) Bezout’s Theorem
Polynomial P(x) is divisible by binomial (x – a) if and only if P(a) = 0.
(iii) Remainder Theorem
If P(x) is divided by (x – a) with remainder r, then P(a) = r.
(iv) Factor Theorem
A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).
(v) The addition, subtraction and multiplication of polynomials P and Q result in a polynomial where,
Degree(P ± Q) ≤ Degree(P or Q)
Degree(P × Q) = Degree(P) + Degree(Q)
(vi) If P(x) = a_{0} + a_{1}x + a_{2}x^{2} + …… + a_{n}x^{n} is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.
The zeros of a polynomial p(x) are all the xvalues that make the polynomial equal to zero.
Let p(x) be a polynomial in x. If p(a) = 0, then we say that 'a' is a zero of the polynomial p(x).
Example 1. Find the zeros of the following linear polynomial
p(x) = 2x + 3
Solution: p(x) = 2x + 3
Now we have to think about the value of x, for which the given function will become zero.
For that let us factor out 2
p(x) = 2 (x + 3/2)
Instead of "x", if we substitute 3/2, p(x) will become zero.
Hence 3/2 is the zero of the given linear polynomial.
Example 2. Find the zeros of the following linear polynomial
Consider, P(x) = 4x + 5 to be a linear polynomial in one variable.
Solution: Let ‘a’ be zero of P(x), then,
P(a) = 4a+5 = 0
Therefore, a = 5/4
In general, if k is zero of the linear polynomial in one variable: P(x) = ax +b, then;
P(k) = ak+b = 0
k = b/a
It can also be written as,
Zero of Polynomial (K) = (Constant/ Coefficient of x)
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