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If one zero of the quadratic polynomial x² + 3x + k is 2, then the value of k is:

-10. Since 2 is a root, P(2) = 0 ⇒ (2)² + 3(2) + k = 0 ⇒ 4 + 6 + k = 0 ⇒ k = -10.

Mathematics · Class 9-10 · CBSE

Polynomials: Zeroes, Graphs and Coefficients

Explore polynomials of degree 1, 2, and 3, geometric meaning of zeroes from Cartesian graphs, algebraic relationship between zeroes and coefficients (sum and product), and constructing quadratic polynomials.

1Degree and Classification of Polynomials

APolynomialin variable x is an algebraic expression involving non-negative integral exponents of x. The highest exponent of x in P(x) is called theDegreeof the polynomial.

Classification of Polynomials by Degree

Type of Polynomial	Degree (n)	General Standard Form	Geometric Graph Shape	Max Real Zeroes
Linear Polynomial	1	P(x) = ax + b (a ≠ 0)	Straight line	$1 \text{zero}\ (at x = -b/a)$
Quadratic Polynomial	2	P(x) = ax² + bx + c (a ≠ 0)	Parabola (opens upward if a > 0, downward if a < 0)	At most 2 real zeroes
Cubic Polynomial	3	P(x) = ax³ + bx² + cx + d (a ≠ 0)	S-shaped curve crossing x-axis	At most 3 real zeroes

Linear Polynomial
Degree (n)
1
General Standard Form
P(x) = ax + b (a ≠ 0)
Geometric Graph Shape
Straight line
Max Real Zeroes
$1 \text{zero}\ (at x = -b/a)$
Quadratic Polynomial
Degree (n)
2
General Standard Form
P(x) = ax² + bx + c (a ≠ 0)
Geometric Graph Shape
Parabola (opens upward if a > 0, downward if a < 0)
Max Real Zeroes
At most 2 real zeroes
Cubic Polynomial
Degree (n)
3
General Standard Form
P(x) = ax³ + bx² + cx + d (a ≠ 0)
Geometric Graph Shape
S-shaped curve crossing x-axis
Max Real Zeroes
At most 3 real zeroes

2Relationship Between Zeroes and Coefficients

If α and β are the zeroes of the quadratic polynomial P(x) = ax² + bx + c, then (x - α) and (x - β) are its factors.

Sum of Zeroes

\alpha + \beta = -\frac{b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}

Product of Zeroes

\alpha \cdot \beta = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2}

Constructing Polynomials from Given Zeroes

Given Information	Quadratic Formula Form	Worked Example
Sum S and Product P are given	P(x) = k [x² - S·x + P]	If S = -3 and P = 2 ⇒ P(x) = x² - (-3)x + 2 = x² + 3x + 2
Two zeroes α and β are given	P(x) = k [x² - (α + β)x + αβ]	If α = 3, β = -2 ⇒ S = 1, P = -6 ⇒ P(x) = x² - x - 6
Symmetric Expressions: α² + β²	(α + β)² - 2αβ	S² - 2P
Symmetric Expressions: 1/α + 1/β	(α + β) / (αβ)	S / P

Sum S and Product P are given
Quadratic Formula Form
P(x) = k [x² - S·x + P]
Worked Example
If S = -3 and P = 2 ⇒ P(x) = x² - (-3)x + 2 = x² + 3x + 2
Two zeroes α and β are given
Quadratic Formula Form
P(x) = k [x² - (α + β)x + αβ]
Worked Example
If α = 3, β = -2 ⇒ S = 1, P = -6 ⇒ P(x) = x² - x - 6
Symmetric Expressions: α² + β²
Quadratic Formula Form
(α + β)² - 2αβ
Worked Example
S² - 2P
Symmetric Expressions: 1/α + 1/β
Quadratic Formula Form
(α + β) / (αβ)
Worked Example
S / P

3Cubic Polynomials and Geometric Roots

Cubic Polynomial Relations: ax³ + bx² + cx + d with zeroes α, β, γ

Relation Type	Formula in Terms of Coefficients	Verification Rule
Sum of Zeroes	α + β + γ = -b/a	- (Coefficient of x²) / (Coefficient of x³)
Sum of Products Taken Two at a Time	αβ + βγ + γα = c/a	(Coefficient of x) / (Coefficient of x³)
Product of Zeroes	α · β · γ = -d/a	- (Constant term) / (Coefficient of x³)

Sum of Zeroes
Formula in Terms of Coefficients
α + β + γ = -b/a
Verification Rule
- (Coefficient of x²) / (Coefficient of x³)
Sum of Products Taken Two at a Time
Formula in Terms of Coefficients
αβ + βγ + γα = c/a
Verification Rule
(Coefficient of x) / (Coefficient of x³)
Product of Zeroes
Formula in Terms of Coefficients
α · β · γ = -d/a
Verification Rule
- (Constant term) / (Coefficient of x³)

4Exam Corner: Tips, Mistakes & Board Questions

Critical Board Exam Graph Takeaways

The number of real zeroes of y = P(x) is strictly equal to the number of points where the graph intersects the x-axis.
If the parabola touches the x-axis at only one point, both roots are real and equal (α = β).
If the parabola lies entirely above or below the x-axis without touching, the polynomial has zero real roots (no real zeroes).

Exam question · CBSE Class 10 Board

Find a quadratic polynomial whose sum and product of zeroes are -1/4 and 1/4 respectively.

A.4x² + x + 1
B.4x² - x + 1
C.x² - 4x + 1
D.4x² + x - 1

Remember

The number of real zeroes is the number ofx-axis intersectionson graph $y = P(x)$ .
For quadratic $ax^2 + bx + c$ : Sum of roots == $\alpha + \beta = -b/a$ ==.
Product of roots == $\alpha\beta = c/a$ ==.
Polynomial formula: == $P(x) = k [x^2 - (\text{Sum})x + (\text{Product})]$ ==.

Check yourself

Question 1
If one zero of the quadratic polynomial x² + 3x + k is 2, then the value of k is: