Examples of Relationship Between Zeroes and Coefficients of Polynomials
Example 1: Find the sum of the roots and the product of the roots of the polynomial x3 -2x2 – x + 2.
Solution:
Given Polynomial,
x3 -2x2 – x + 2
comparing with ax3 + bx2 + cx + d = 0
a = 1, b= -2, c = -1, and d = 2
Sum of the roots (p + q+ r) = – Coefficient of x2/ coefficient of x3
= -b/a
= -(-2)/1 = 2Product of the roots (pqr) = – Constant Term/Coefficient of x3
= -d/a
= -2/1 = -2
Example 2: Find the sum and product of the zeros of the quadratic polynomial 6x2 + 18 = 0
Solution:
Given Polynomial 6x2 + 18 = 0
It can be also written as, 6x2 + 0x + 18 = 0
Comparing with ax2 + bx + c = 0
a = 6, b = 0, and c = 18
Sum of Zeroes = – Coefficient of x/ Coefficient of x2
= -b/a
= -0/6
= 0Product of the Zeroes = Constant term / Coefficient of x2
= c/a
= 18/6
= 3
Example 3: For the given polynomial ax2 + bx + 1 = 0. Its roots are -1 and 3. Find the values of a and b.
Solution:
Let m and n be the roots of the quadratic equation ax2 + bx + 1 = 0
Here,
- m = -1
- n = 3
We know that,
m + n = -b/a
⇒ -1 + 3 = -b/a
⇒ -b/a = 2…(i)
And m.n = c/a
⇒ (-1)(3) = 1/a
⇒ -3 = 1/a
⇒ a = -1/3…(ii)
from (i) we get,
-b/a = 2
⇒ b = -2a
⇒ b = -2(-1/3) = 2/3
Relationship between Zeroes and Coefficients of a Polynomial
Polynomials are algebraic expressions with constants and variables that can be linear i.e. the highest power o the variable is one, quadratic and others. The zeros of the polynomials are the values of the variable (say x) that on substituting in the polynomial give the answer as zero.
While the coefficients of a polynomial are the constants that are multiplied by the variables of the polynomial. There is a relation between the Zeroes of a Polynomial and the Coefficients of a Polynomial which is widely used in solving problems in algebra.
In this article, we will learn about the Zeroes of a Polynomial, the Coefficients of a Polynomial, and their relation in detail.
Table of Content
- Zeroes of a Polynomial
- Coefficients of a Polynomial
- Relationship between Zeros and Coefficients of a Polynomial
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial
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