MathematicsUnit 86 min read

Polynomial Equations: Roots, Factorization & Graphs

Unit 8 of Mathematics covers solving polynomial equations (linear, quadratic, cubic), finding roots using factorization, synthetic division, and graphical methods, and applying these to real-world problems.

Padhai · Based on the official syllabus and past papers
Updated 26 Sept 2026

TAKEAWAYS:

Learn to solve linear, quadratic, and cubic polynomial equations using factorization, formula, and synthetic division.
Understand how roots of a polynomial relate to its factors and graph.
Use graphical methods to estimate roots when exact solutions are difficult.
Apply polynomial equations to real-world problems like area, profit, and motion.
Master synthetic division to check roots and factorize polynomials efficiently.

1. What is a Polynomial Equation?

A polynomial equation is an equation of the form: $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0} = 0$ where $a_{n}, a_{n - 1}, \dots, a_{0}$ are constants, and $n$ is a non-negative integer.

Types of Polynomial Equations

Degree	Name	Example
1	Linear	$2 x + 3 = 0$
2	Quadratic	$x^{2} - 5 x + 6 = 0$
3	Cubic	$x^{3} - 6 x^{2} + 11 x - 6 = 0$
4+	Higher degree	$x^{4} - 5 x^{2} + 4 = 0$

Key Terms

Root (Zero): A value $x = α$ such that $P (α) = 0$ .
Factor: An expression $(x - α)$ such that $P (x) = (x - α) Q (x)$ .
Multiplicity: How many times a root repeats (e.g., $(x - 2)^{3}$ has root $x = 2$ with multiplicity 3).

2. Solving Linear Polynomial Equations

A linear polynomial has the form: $a x + b = 0$ Solution: $x = - \frac{b}{a}$

Example 1: Solve $3 x - 7 = 0$ . Solution: $3 x = 7$ $x = \frac{7}{3}$

3. Solving Quadratic Polynomial Equations

A quadratic polynomial has the form: $a x^{2} + b x + c = 0$ Methods to solve:

Factorization (if possible)
Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Completing the square

Example 2: Solve $x^{2} - 5 x + 6 = 0$ by factorization.

Solution: Find two numbers that multiply to $6$ and add to $- 5$ . These numbers are $- 2$ and $- 3$ . So, $x^{2} - 5 x + 6 = (x - 2) (x - 3) = 0$ Thus, roots are: $x = 2 or x = 3$

Example 3: Solve $2 x^{2} - 4 x - 6 = 0$ using the quadratic formula.

Solution: Here, $a = 2$ , $b = - 4$ , $c = - 6$ . $x = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( 2 ) ( - 6 )}{2 ( 2 )}$ $x = \frac{4 \pm 16 + 48}{4}$ $x = \frac{4 \pm 64}{4}$ $x = \frac{4 \pm 8}{4}$ Thus, $x = \frac{12}{4} = 3 or x = \frac{- 4}{4} = - 1$

4. Solving Cubic Polynomial Equations

A cubic polynomial has the form: $a x^{3} + b x^{2} + c x + d = 0$ Methods:

Factorization (if one root is obvious)
Synthetic division (to reduce degree after finding one root)
Cardano’s formula (for complex cases, beyond NEB scope)

Example 4: Solve $x^{3} - 6 x^{2} + 11 x - 6 = 0$ .

Step 1: Try possible rational roots (factors of 6: $\pm 1, \pm 2, \pm 3, \pm 6$ ). Let $x = 1$ : $1 - 6 + 11 - 6 = 0$ So, $x = 1$ is a root.

Step 2: Use synthetic division to factor out $(x - 1)$ .

After division, we get: $x^{3} - 6 x^{2} + 11 x - 6 = (x - 1) (x^{2} - 5 x + 6)$

Step 3: Factor the quadratic: $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ Thus, the roots are: $x = 1, x = 2, x = 3$

5. Graphical Method for Polynomial Equations

The graph of a polynomial crosses the x-axis at its roots.

Even multiplicity roots touch but do not cross the x-axis.
Odd multiplicity roots cross the x-axis.

Example 5: Sketch the graph of $y = (x - 1)^{2} (x + 2)$ .

Roots:

$x = 1$ (multiplicity 2, touches but does not cross)
$x = - 2$ (multiplicity 1, crosses)

6. Applications of Polynomial Equations

Polynomials are used in:

Area and Volume Problems
Profit and Loss Calculations
Physics (Motion, Projectile Problems)
Economics (Cost, Revenue, Break-even Points)

Example 6: A rectangular garden has an area of 120 m². If its length is 3 m more than its width, find its dimensions.

Solution: Let width = $x$ m. Then, length = $x + 3$ m. Area = length × width: $x (x + 3) = 120$ $x^{2} + 3 x - 120 = 0$ Using the quadratic formula: $x = \frac{- 3 \pm 9 + 480}{2}$ $x = \frac{- 3 \pm 489}{2}$ Since width cannot be negative: $x = \frac{- 3 + 489}{2} \approx 10.3 m$ Length ≈ 13.3 m.

7. Summary Table: Methods for Solving Polynomial Equations

Type	Method	When to Use
Linear	Direct solution	Always applicable
Quadratic	Factorization	If easily factorable
Quadratic	Quadratic formula	Always applicable
Cubic	Factorization + Synthetic Division	If one root is obvious
Higher Degree	Graphical estimation	When exact solutions are complex

Exam Tip

✅ For NEB exams, focus on:

Factorization (most common in NEB questions).
Quadratic formula (always works for quadratics).
Synthetic division (useful for cubics).
Graphical interpretation (roots correspond to x-intercepts).
Real-world applications (area, profit, motion problems).

❌ Avoid memorizing formulas—understand the logic behind solving polynomials.

NEB Board-Style Questions (Practice)

Solve the following equations: a) $3 x^{2} - 10 x + 7 = 0$ b) $x^{3} - 5 x^{2} + 6 x = 0$ c) $2 x^{2} - 5 x + 3 = 0$ (using the quadratic formula)
If $P (x) = x^{3} - 4 x^{2} + x + 6$ , show that $x = 2$ is a root and factorize $P (x)$ .
A rectangular field has an area of 200 m². If its length is 5 m more than its width, find its dimensions.
Sketch the graph of $y = (x + 1) (x - 2)^{2}$ and state the nature of its roots.

Final Note: Practice solving polynomials by factorization first. If stuck, use the quadratic formula or synthetic division. Always verify your roots by substitution! 🚀

Based on the NEB +2 Science syllabus for Mathematics (Maths), unit 8.

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