CBSE Class 10th CBSE Class 10th Relationship between Zeroes and Coefficients of a Polynomial Details & Preparations Downloads

In the enchanting realm of Class 10 mathematics, the study of polynomials unveils a captivating connection between their zeroes and coefficients. This intricate relationship not only deepens our understanding of algebra but also provides valuable insights into the behavior of these mathematical expressions. Let's embark on a journey to explore the nuanced relationship between the zeroes and coefficients of a polynomial.

Harmony in Equations: Unveiling the Class 10 Enigma - Exploring the Dynamic Relationship between Polynomial Zeroes and Coefficients

Relationship Between the Zeros and Coefficients of a Polynomial

A real number say “a” is a zero of a polynomial P(x) if P(a) = 0. The zero of a polynomial is clearly explained using the Factor theorem. If “k” is a zero of a polynomial P(x), then (x-k) is a factor of a given polynomial. The relation between the zeros and the coefficients of a polynomial is given below

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)

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 ax² + 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 x²

The product of zeros, αβ is c/a = Constant term / Coefficient of x²

Cubic Polynomial

The cubic polynomial is a polynomial with the highest degree of 3. The cubic polynomial should be in the form of ax3 + bx2 + 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 x2/ coefficient of x3

The sum of the product of zeros, αβ+ βγ + αγ is c/a = Coefficient of x/Coefficient of x3

The product of zeros, αβγ is -d/a = – Constant term/Coefficient of x3

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Zeros of a Polynomial Solved Examples

Example Evaluate the sum and product of zeros of the quadratic polynomial 4x² – 9.

Solution

Given quadratic polynomial is 4x2 – 9.

 4x2 – 9 can be written as 2x2 – 33, which is equal to (2x+3)(2x-3).

To find the zeros of a polynomial, equate the above expression to 0

(2x+3)(2x-3) = 0

2x+3 = 0

2x = -3

X = -3/2

Similarly, 2x-3 = 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.

Viète's Formulas

François Viète, a French mathematician from the 16th century, provided us with elegant formulas that express the relationship between the zeroes and coefficients of a polynomial. For a quadratic polynomial.

Practical Applications:

Understanding this relationship has practical implications. By knowing the sum and product of zeroes, one can deduce information about the coefficients, aiding in solving equations and analyzing the behavior of polynomials.

CBSE Class 10th Downloadable Resources: 

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SAMPLE PRACTICE QUESTION

Q1: What is the geometrical meaning of the zeroes of a polynomial?

Ans: The zeroes of a polynomial represent the x-values at which the polynomial intersects the x-axis on a graph, indicating points where the polynomial evaluates to zero.

Q2: How do you visually interpret the zeroes on a graph?

Ans: Zeroes correspond to the x-coordinates where the polynomial function crosses or touches the x-axis, providing key information about the roots.

Q3: Can a polynomial have complex or imaginary zeroes with no real intersection on the graph?

Ans: Yes, a polynomial can have complex or imaginary zeroes that don't intersect the real axis on the graph, but they still contribute to the overall understanding of the polynomial's behavior.

Q4: How do multiple zeroes of a polynomial affect the shape of its graph?

Ans: The number of zeroes influences the graph's behavior. Double zeroes may create points of inflection, while triple zeroes may result in a flattened curve at the x-axis.

Q5: Do all polynomial graphs intersect the x-axis at the zeroes?

Ans: Not necessarily. The graph may touch the x-axis at a zero (with multiplicity one) or cross it (with odd multiplicity), influencing the graph's behavior around those points.

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