How to Solve Polynomial Equations

How to Solve Polynomial Equations

(For SSC / Bank / Railway Exams – Ace Your Math Section!)

Introduction

"Master polynomial equations, and you unlock 5–10 marks in your SSC/Bank/Railway exam—questions that look scary but follow a simple 3-step method. Let’s break it down."

(On camera: Hold up a past paper with a polynomial question highlighted.) "This one question could be the difference between a pass and a fail. Today, you’ll learn how to solve it in under 2 minutes."

What You Need To Know First

Before diving in, ensure you understand: 1. Factorisation – Breaking expressions like (x^2 - 5x + 6) into ((x-2)(x-3)). 2. Zero Product Property – If (A \times B = 0), then (A = 0) or (B = 0). 3. Basic Algebra – Solving linear equations like (2x + 3 = 0).

(On camera: Quick quiz – "Can you factorise (x^2 - 9)? If yes, you’re ready!)

Key Vocabulary

(On camera: Point to each term and give a 2-second example.)

Formulas To Know

1. Quadratic Formula (for (ax^2 + bx + c = 0))
   [
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   ]
2. (a, b, c) = coefficients of (x^2, x, \text{constant}).

3. MEMORISE THIS (not always given in exams).

4. Sum & Product of Roots (for (ax^2 + bx + c = 0))
   [
   \text{Sum} = -\frac{b}{a}, \quad \text{Product} = \frac{c}{a}
   ]

5. "Given on exam sheet" (but memorising saves time).

6. Factor Theorem

7. If (f(a) = 0), then ((x - a)) is a factor of (f(x)).
8. MEMORISE THIS (critical for cubic/quartic equations).

(On camera: Write each formula on screen, then cover and ask: "What’s the quadratic formula?")

Step-by-Step Method

Goal: Solve (P(x) = 0) (find all roots).

Step 1: Check for Common Factors

• Factor out the greatest common factor (GCF) first.
• Example: (2x^2 + 4x = 0) → (2x(x + 2) = 0).

Step 2: Factorise (If Possible)

• Quadratic (Degree 2): Use factorisation, completing the square, or quadratic formula.
• Example: (x^2 - 5x + 6 = (x-2)(x-3)).
• Cubic (Degree 3): Use Factor Theorem to guess a root, then factorise.
• Example: (x^3 - 6x^2 + 11x - 6 = 0) → Try (x=1) → ((x-1)(x^2 -5x +6)).

Step 3: Apply Zero Product Property

• Set each factor = 0 and solve.
• Example: ((x-2)(x-3) = 0) → (x = 2) or (x = 3).

Step 4: Check for Extraneous Solutions

• If you squared both sides (e.g., in equations with roots), verify solutions in the original equation.

(On camera: Demonstrate each step with a simple example, like (x^2 - 9 = 0).)

Worked Examples

Example 1 – Basic (Quadratic)

Solve: (x^2 - 5x + 6 = 0)

Step 1: No common factors. Step 2: Factorise → ((x-2)(x-3) = 0). Step 3: Zero Product Property → (x = 2) or (x = 3). Answer: (x = 2, 3).

What we did and why: - We factored the quadratic because it’s the fastest method when possible. - Always check if the quadratic can be factored before using the formula.

Example 2 – Medium (Cubic)

Solve: (x^3 - 6x^2 + 11x - 6 = 0)

Step 1: No common factors. Step 2: Use Factor Theorem → Try (x=1):
(1 - 6 + 11 - 6 = 0) → ((x-1)) is a factor.
Divide (x^3 - 6x^2 + 11x - 6) by ((x-1)) → (x^2 -5x +6). Step 3: Factorise (x^2 -5x +6) → ((x-2)(x-3)).
So, (x^3 - 6x^2 + 11x - 6 = (x-1)(x-2)(x-3)). Step 4: Zero Product Property → (x = 1, 2, 3). Answer: (x = 1, 2, 3).

What we did and why: - We guessed (x=1) because it’s a common root for simple cubics. - Always divide the polynomial by ((x - \text{root})) to reduce the degree.

Example 3 – Exam-Style (Disguised Quadratic)

Solve: (2x^4 - 5x^2 + 3 = 0)

Step 1: Let (y = x^2) → Equation becomes (2y^2 -5y +3 = 0). Step 2: Factorise → ((2y-3)(y-1) = 0). Step 3: Solve for (y) → (y = \frac{3}{2}) or (y = 1). Step 4: Substitute back (y = x^2) →
- (x^2 = \frac{3}{2}) → (x = \pm \sqrt{\frac{3}{2}}).
- (x^2 = 1) → (x = \pm 1). Answer: (x = \pm 1, \pm \sqrt{\frac{3}{2}}).

What we did and why: - We used substitution to turn a quartic into a quadratic. - Always check if the equation can be simplified with substitution.

Common Mistakes

(On camera: Hold up a wrong solution and ask: "Spot the mistake!)

Exam Traps

(On camera: Show a past paper question with a trap and say: "This is a classic! Here’s how to beat it.")

1-Minute Recap

(Spoken naturally, as if to a student the night before the exam.)

"Okay, listen up—this is all you need to remember for polynomial equations:

1. Always factor out the GCF first. If there’s a common term, take it out.
2. For quadratics: Try factorising first. If it doesn’t work, use the quadratic formula. Memorise it—it’s your lifeline.
3. For cubics: Guess a root (try (x=1, -1, 2, -2)), then factorise using the Factor Theorem.
4. If it’s a quartic with even powers (like (x^4)), substitute (y = x^2) to turn it into a quadratic.
5. Never forget the Zero Product Property: If (A \times B = 0), then (A=0) or (B=0). This is how you find the roots!

And one last thing—always check your answers. Plug them back into the original equation to make sure they work. Examiners love to trick you with extraneous roots.

You’ve got this. Now go solve those equations like a pro!