Factorisation & Introduction to Graphs — Class 8

Combined chapter notes: Factorisation methods and an easy introduction to Graphs — SEO-friendly, mobile-first layout with dark-mode toggle and printable pages.

1. Why Factorise?

Factorisation writes an expression as a product of simpler expressions (factors). It's useful for solving equations, simplifying expressions, and finding roots. Think of factorisation as 'reverse expansion'.

Key idea: converting sums into products often reveals structure and simplifies solving problems.

2. Methods of Factorisation (Quick Guide)

A. Highest Common Factor (HCF / GCF)

Always check for a common factor first. Factor it out to simplify remaining work.

Example: 12x^2 − 8x = 4x(3x − 2).
B. Factor by Grouping

Group terms to create common factors. Useful for 4-term expressions or after splitting middle term in quadratics.

Example: ax + ay + bx + by = (a+b)(x+y).
C. Split Middle Term (Quadratics)

For ax^2 + bx + c, find numbers p,q such that p+q=b and pq=ac, then group and factor.

Example: x^2 + 5x + 6 = (x+2)(x+3).
D. Use Standard Identities

Recognise patterns (difference of squares, sum/difference of cubes, perfect squares) to factor instantly.

E. Special Cases

Perfect squares and sum/difference of cubes often appear—memorize their forms for speed.

3. Useful Identities (Factorisation)

  • a^2 − b^2 = (a − b)(a + b) — Difference of squares
  • a^2 + 2ab + b^2 = (a + b)^2 and a^2 − 2ab + b^2 = (a − b)^2
  • a^3 + b^3 = (a + b)(a^2 − ab + b^2)
  • a^3 − b^3 = (a − b)(a^2 + ab + b^2)

4. Worked Examples — Factorisation

Example 1: Factorise 3x^2 + 11x + 6.
ac = 18. Find 9 and 2 → 3x^2 + 9x + 2x + 6 = 3x(x+3)+2(x+3)=(3x+2)(x+3).
Example 2: Factorise x^3 − 8.
Difference of cubes: (x−2)(x^2+2x+4).
Example 3: Factorise 6x^2 − x − 2.
ac = −12. Numbers 3 and −4 give −1 → split and group → (3x+2)(2x−1).

5. Introduction to Graphs

Graphs visually represent numerical information. They help compare quantities, show trends over time, and display relationships between variables.

Common contexts: comparing sales across months, showing temperature changes, or plotting distance vs time.

6. Types of Graphs

  • Bar Graphs: Compare discrete categories (e.g., students in classes). Bars should have equal width and gaps.
  • Line Graphs: Show trends over time (e.g., months vs sales). Plot points and join with lines.
  • Pie Charts: Show percentage parts of a whole. Convert frequencies to angles: (freq/total)×360°.
  • Histogram: For continuous grouped data — bars touch; area represents frequency.
  • Cartesian (x–y) Graphs: Plot points (x,y) on coordinate plane to show relationships between two variables.

7. Plotting & Reading Graphs (Basics)

  1. Choose axes: x-axis (independent) and y-axis (dependent). Label units clearly.
  2. Scale: Choose a suitable, even scale so data fits neatly. Write scale on axes.
  3. Plot points: For (x,y), move x units on x-axis, then y units up (or down if negative).
  4. Draw/Connect: For line graphs join points smoothly; for bar graphs draw bars to correct height.
  5. Interpret: Read values by tracing horizontally/vertically to axes. Identify maxima/minima/trends.
Example (reading): If a line graph shows distance (km) on y and time (h) on x, slope gives speed (km/h). A steeper slope → higher speed.

8. Practice: Small Activities

  1. Make a bar graph: number of books read by 5 students: {A:3,B:5,C:2,D:4,E:6}.
  2. Plot a line graph: temperature over 7 days: {30,31,28,29,33,35,34} and observe trend.
  3. Create a pie chart: marks out of 100 in subjects and convert to angles.

Sketch on paper or use spreadsheet tools (Excel/Google Sheets) for cleaner charts.