Table of contents  
What are Polynomials?  
Degree of a Polynomial  
Zeros of Polynomial  
Relationships between Zeros and Coefficient of Polynomials 
The zeroes of a polynomial p(x) are precisely the xcoordinates of the points, where the graph of y = p(x) intersects the xaxis.
For example:
Let's consider the polynomial f(x) = x^{2}  4.
1. Linear Polynomial
The linear polynomial is an expression, in which the degree of the polynomial is 1. The linear polynomial should be in the form of ax+b. Here, “x” is a variable, “a” and “b” are constant.
The polynomial P(x) is ax+b, then the zero of a polynomial is b/a = – constant term/coefficient of x
For Example: P(x)=2x+4
Coefficient of x (i.e., "a") is 2. Constant Term is 4.
The zero is given by:
x =  constant term/ coefficient of x =  4/2 = 2
Thus, the zero pf P(x)= 2x + 4 is x = 2. This can be verifies by substituting x = 2 into the polynomial:
P(2) = 2(2) + 4 = 4 + 4 = 0
So, x = 2 is the zero of the polynomial P(x)= 2x + 4.
2. Quadratic Polynomial
The Quadratic polynomial is defined as a polynomial with the highest degree of 2. The quadratic polynomial should be in the form of ax^{2} + bx + c. In this case, a ≠ 0. Let say α and β are the two zeros of a polynomial, then
The sum of zeros, α + β is b/a = – Coefficient of x/ Coefficient of x^{2}
The product of zeros, αβ is c/a = Constant term / Coefficient of x^{2}
For Example: Evaluate the sum and product of zeros of the quadratic polynomial 4x^{2}^{ }– 9.
Solution: Given quadratic polynomial is 4x^{2 }– 9.
4x^{2} – 9 can be written as 2x^{2} – 3^{3}, which is equal to (2x+3)(2x3).
To find the zeros of a polynomial, equate the above expression to 0
(2x+3)(2x3) = 0
2x+3 = 0
2x = 3
x = 3/2
Similarly, 2x3 = 0,
2x = 3
x = 3/2
Therefore, the zeros of a given quadratic polynomial is 3/2 and 3/2.
Finding the sum and product of a polynomial:
The sum of the zeros = (3/2)+ (3/2) = (3/2)(3/2) = 0
The product of zeros = (3/2).(3/2) = 9/4.
3. Cubic Polynomial
The cubic polynomial is a polynomial with the highest degree of 3. The cubic polynomial should be in the form of ax^{3} + bx^{2} + cx + d, where a ≠ 0. Let say α, β, and γ are the three zeros of a polynomial, then
The sum of zeros, α + β + γ is b/a = – Coefficient of x^{2}/ coefficient of x^{3}
The sum of the product of zeros, αβ+ βγ + αγ is c/a = Coefficient of x/Coefficient of x^{3}
The product of zeros, αβγ is d/a = – Constant term/Coefficient of x^{3}
For Example: Find the sum of the roots and the product of the roots of the polynomial x^{3} 2x^{2} – x + 2.
Solution: Given Polynomial, x^{3} 2x^{2} – x + 2comparing with ax^{3} + bx^{2} + cx + d = 0
a = 1, b= 2, c = 1, and d = 2
Sum of the roots (p + q+ r) = – Coefficient of x^{2}/ coefficient of x^{3}
= b/a
= (2)/1 = 2Product of the roots (pqr) = – Constant Term/Coefficient of x^{3}
= d/a
= 2/1 = 2$P(x)\; =\; 2x^3\; \; 3x^2\; +\; 4x\; \; 5$
116 videos420 docs77 tests

1. What are polynomials and how are they defined? 
2. How is the degree of a polynomial determined? 
3. What are zeros of a polynomial and how are they related to the polynomial's factors? 
4. How can we find the relationship between the zeros and coefficients of a polynomial? 
5. What is the Division Algorithm for polynomials and how is it used? 

Explore Courses for Class 10 exam
