CBSE Class 10 Maths Notes: Polynomials

📚 Definitions and Core Concepts

Let’s dive into the world of Polynomials! Here, we’ll explore fundamental definitions and concepts.

Definition and Degree of a Polynomial

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
The degree of a polynomial is the highest power of the variable in the polynomial.

• Examples:
• $2x^2 + 3x – 1$ (degree 2, quadratic)
• $5x^3 – 4x + 2$ (degree 3, cubic)
• $7x + 1$ (degree 1, linear)

Coefficients and Constant Term

In a polynomial, the coefficients are the numerical values that multiply the variables. The constant term is the term without any variable (the term with $x^0$).

• Examples:
• In $3x^2 – 5x + 2$, the coefficients are $3, -5$ and the constant term is $2$.

🔎 Zeros (Roots) of a Polynomial

Understanding the zeros is crucial for solving and analyzing polynomials.

Graphical Interpretation

A zero (or root) of a polynomial is a value of the variable for which the polynomial equals zero. Graphically, the zeros are the x-coordinates of the points where the graph of the polynomial intersects the x-axis.

• A linear polynomial ($ax + b$) will intersect the x-axis at one point.
• A quadratic polynomial ($ax^2 + bx + c$) can intersect the x-axis at zero, one, or two points.

Algebraic Interpretation

Algebraically, a zero $x = \alpha$ satisfies the equation $p(\alpha) = 0$, where $p(x)$ is the polynomial.

🤝 Relationship Between Zeros and Coefficients (Quadratic Polynomials)

There exists a clear relationship between the zeros and the coefficients of a quadratic polynomial.

Sum and Product of Roots

For a quadratic polynomial of the form $ax^2 + bx + c$, where $a \ne 0$:

• The sum of the zeros (roots) is: $-\frac{b}{a}$
• The product of the zeros (roots) is: $\frac{c}{a}$

💡 Finding Zeros and Verifying Relations

Learn methods to find the zeros and verify the relationships.

Finding Zeros by Factorisation and Inspection

• Factorisation: Factoring the polynomial allows you to find the values of $x$ that make each factor equal to zero.
• By Inspection: In simple cases, you can identify the zeros directly by observing the values that satisfy the polynomial.

Verifying Relations Using Examples

• Example 1: Consider the polynomial $x^2 – 5x + 6$.
• Factorizing, we get $(x – 2)(x – 3)$.
• Therefore, the zeros are $2$ and $3$.
• Sum of zeros: $2 + 3 = 5$ and $-\frac{b}{a} = -\frac{-5}{1} = 5$.
• Product of zeros: $2 \times 3 = 6$ and $\frac{c}{a} = \frac{6}{1} = 6$.
• Example 2: Consider the polynomial $2x^2 + 7x + 3$
• Factorizing: $(2x + 1)(x + 3)$
• Zeros are $-\frac{1}{2}$ and $-3$
• Sum of Zeros: $-\frac{1}{2} + (-3) = -\frac{7}{2}$ and $-\frac{b}{a} = -\frac{7}{2}$
• Product of Zeros: $-\frac{1}{2} \times (-3) = \frac{3}{2}$ and $\frac{c}{a} = \frac{3}{2}$