What is Solving Polynomial Equations by Factorisation?

S3-SA1-0657

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Solving polynomial equations by factorisation means finding the values of the variable (like 'x') that make the equation true, by breaking down the polynomial into simpler multiplication parts (factors). It's like finding the secret numbers that, when multiplied, give you the original equation, and then using those secrets to find 'x'.

Simple Example

Quick Example

Imagine you have a rectangular cricket ground, and its area is 24 square meters. If the length is 'x + 2' meters and the width is 'x - 2' meters, finding 'x' using factorisation helps you figure out the exact length and width. You'd set up the equation (x + 2)(x - 2) = 24 and solve for 'x'.

Worked Example

Step-by-Step

Let's solve the equation: x^2 - 5x + 6 = 0

1. Identify the polynomial: We have a quadratic polynomial, x^2 - 5x + 6.
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2. Find two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of 'x'). The numbers are -2 and -3.
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3. Rewrite the middle term (-5x) using these numbers: x^2 - 2x - 3x + 6 = 0.
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4. Group the terms and factor out common factors: (x^2 - 2x) + (-3x + 6) = 0. This becomes x(x - 2) - 3(x - 2) = 0.
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5. Factor out the common bracket (x - 2): (x - 2)(x - 3) = 0.
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6. Set each factor equal to zero and solve for x: x - 2 = 0 OR x - 3 = 0.
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7. Solve for x: x = 2 OR x = 3.
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Answer: The solutions (roots) of the equation are x = 2 and x = 3.

Why It Matters

Understanding factorisation is super important in many advanced fields! Engineers use it to design bridges and buildings, while computer scientists use it in coding and game development. Even data scientists use similar logic to understand patterns in large datasets, helping companies make smart decisions.

Common Mistakes

MISTAKE: Not setting the equation to zero before factorising. | CORRECTION: Always rearrange the polynomial equation so that one side is zero (e.g., ax^2 + bx + c = 0) before you start factorising.

MISTAKE: Incorrectly finding the two numbers for splitting the middle term. For x^2 + bx + c, students might look for numbers that add to 'c' and multiply to 'b'. | CORRECTION: The two numbers must MULTIPLY to 'c' and ADD to 'b'.

MISTAKE: Forgetting to set EACH factor to zero after factorising. For example, after (x-2)(x-3)=0, some students just write x=2. | CORRECTION: Remember that if the product of two factors is zero, at least one of them must be zero. So, set (x-2)=0 AND (x-3)=0 to find all possible solutions.

Practice Questions

Try It Yourself

QUESTION: Solve the equation by factorisation: x^2 - 7x + 10 = 0 | ANSWER: x = 2, x = 5

QUESTION: Find the solutions for the equation: y^2 + 8y + 15 = 0 | ANSWER: y = -3, y = -5

QUESTION: Solve for 'a': 2a^2 + 5a - 3 = 0 (Hint: Multiply the coefficient of a^2 and the constant term first) | ANSWER: a = 1/2, a = -3

MCQ

Quick Quiz

Which of the following is a factorised form of x^2 + 6x + 9?

(x + 3)(x - 3)

(x + 3)^2

(x + 6)(x + 3)

(x - 3)^2

The Correct Answer Is:

For x^2 + 6x + 9, we need two numbers that multiply to 9 and add to 6. These numbers are 3 and 3. So, the factors are (x + 3)(x + 3), which is (x + 3)^2.

Real World Connection

In the Real World

Imagine you're designing a new mobile game level in India. You might use polynomial equations to model the trajectory of a character jumping or the path of a cricket ball. Solving these equations by factorisation helps game developers predict exactly where things will land, making the game realistic and fun.

Key Vocabulary

Key Terms

Polynomial: An expression with one or more terms, where each term has variables raised to non-negative integer powers.| Factorisation: The process of breaking down an expression into a product of simpler expressions (factors).| Root/Solution: The value(s) of the variable that make the equation true.| Quadratic Equation: A polynomial equation of degree 2 (highest power of the variable is 2).

What's Next

What to Learn Next

Great job understanding factorisation! Next, you can explore 'Solving Quadratic Equations using the Quadratic Formula'. This method helps you solve polynomial equations that might be tricky to factorise, giving you another powerful tool for finding solutions.