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CBSE Class 10 Maths Chapter 2: Polynomials Notes

📖 Chapter Notes ✏️ NCERT Solutions 📥 PDF Notes

1. What is a Polynomial?

An algebraic expression where the exponents of the variables are strictly **whole numbers**. The highest power of the variable in a polynomial is called its degree.

2. Geometrical Meaning of Zeroes

The number of zeroes of a polynomial $p(x)$ is exactly equal to the number of times its graph intersects or touches the X-axis.

3. Relationship Between Zeroes and Coefficients

If $\alpha$ and $\beta$ are the zeroes of a quadratic polynomial $p(x) = ax^2 + bx + c$, then:

👉 Sum of Zeroes ($\alpha + \beta$) = $-b/a = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$

👉 Product of Zeroes ($\alpha \cdot \beta$) = $c/a = \frac{\text{Constant term}}{\text{Coefficient of } x^2}$
CBSE Board Favorite Question

Forming a Polynomial: If sum and product of zeroes are given, the quadratic polynomial is written as:
$$p(x) = k[x^2 - (\text{Sum of Zeroes})x + (\text{Product of Zeroes})]$$

✏️ Complete NCERT Solutions Class 10 Polynomials

Exercise 2.1
Q1. The graphs of $y = p(x)$ are given. Find the number of zeroes of $p(x)$ in each case.
Rule: Count how many times the line crosses the horizontal X-axis line.
1. If the graph is parallel to the X-axis (does not touch it): 0 zeroes.
2. If the graph intersects the X-axis at exactly one point: 1 zero.
3. If it intersects at three points: 3 zeroes.
Exercise 2.2
Q1. Find the zeroes of the quadratic polynomial $x^2 - 2x - 8$ and verify the relationship with coefficients.
Step 1: Factorize by splitting middle term
$x^2 - 4x + 2x - 8 = 0$
$x(x - 4) + 2(x - 4) = 0$
$(x - 4)(x + 2) = 0 \implies \mathbf{\alpha = 4, \beta = -2}$
Step 2: Verify Sum
$\alpha + \beta = 4 + (-2) = 2$
From equation, $-b/a = -(-2)/1 = 2$. (Verified!)
Step 3: Verify Product
$\alpha \cdot \beta = 4 \times (-2) = -8$
From equation, $c/a = -8/1 = -8$. (Verified!)