\displaystyle \textbf{Chapter 5: Factorisation - Concept Notes}

\displaystyle \textbf{1. Introduction}
\displaystyle \text{Factorisation is the process of expressing an algebraic expression as the product} \\ \text{of two or more expressions called factors.}
\displaystyle \text{Example: }x^2+5x+6=(x+3)(x+2)
\displaystyle \text{Here, }(x+3)\text{ and }(x+2)\text{ are the factors of }x^2+5x+6.
\displaystyle \textbf{Important Fact: Factorisation is the reverse of multiplication.}
\displaystyle \\

\displaystyle \textbf{2. Methods of Factorisation}
\displaystyle \text{The chapter discusses the following five methods of factorisation.}
\displaystyle \\

\displaystyle \textbf{Method 1: Taking Out the Common Factor}
\displaystyle \textbf{When to Use: }\text{Every term of the expression has a common factor.}
\displaystyle \textbf{Procedure:}
\displaystyle \text{Step 1: Find the H.C.F. of all the terms.}
\displaystyle \text{Step 2: Take the H.C.F. outside the bracket.}
\displaystyle \text{Step 3: Write the remaining quotient inside the bracket.}
\displaystyle \text{Example: }6a^2-3ax=3a(2a-x)
\displaystyle \textbf{The common factor may be a number, a variable or an algebraic expression.}
\displaystyle \text{Examples: }12x+18=6(2x+3),\qquad 8ab^2+12a^2b=4ab(2b+3a)
\displaystyle \text{Example with algebraic expression: }4(x+y)^2-3(x+y)=(x+y)[4(x+y)-3]
\displaystyle \\

\displaystyle \textbf{Method 2: Grouping Method}
\displaystyle \textbf{When to Use: }\text{Expressions containing four or more terms that can be grouped} \\ \text{suitably.}
\displaystyle \textbf{Procedure:}
\displaystyle \text{Step 1: Arrange the terms into suitable groups.}
\displaystyle \text{Step 2: Factorise each group separately.}
\displaystyle \text{Step 3: Take out the common binomial factor.}
\displaystyle \text{Example: }ab+bc+ax+cx=(ab+bc)+(ax+cx)
\displaystyle =b(a+c)+x(a+c)=(a+c)(b+x)
\displaystyle \text{Sometimes rearranging the terms first makes grouping possible.}
\displaystyle \\

\displaystyle \textbf{Method 3: Factorisation of a Trinomial }ax^2+bx+c\textbf{ (By Splitting the Middle} \\ \text{Term)}
\displaystyle \textbf{When to Use: }\text{Quadratic expressions having three terms.}
\displaystyle \textbf{Procedure:}
\displaystyle \text{Step 1: Calculate }ac.
\displaystyle \text{Step 2: Find two numbers whose sum is }b\text{ and product is }ac.
\displaystyle \text{Step 3: Split the middle term using these two numbers.}
\displaystyle \text{Step 4: Apply the grouping method.}
\displaystyle \text{Example: }x^2+5x+6=x^2+3x+2x+6=x(x+3)+2(x+3)=(x+3)(x+2)
\displaystyle \text{Example: }2x^2-7x+6=2x^2-4x-3x+6=2x(x-2)-3(x-2)=(x-2)(2x-3)
\displaystyle \textbf{Checking Factorisability:}
\displaystyle \text{For }ax^2+bx+c,\text{ calculate }b^2-4ac.
\displaystyle \text{If }b^2-4ac\text{ is a perfect square, the quadratic is factorisable; otherwise it is not} \\ \text{factorisable.}
\displaystyle \\

\displaystyle \textbf{Method 4: Difference of Two Squares}
\displaystyle \textbf{Identity: }a^2-b^2=(a+b)(a-b)
\displaystyle \textbf{When to Use: }\text{The expression is the difference of two perfect squares.}
\displaystyle \text{Example: }x^2-25=(x+5)(x-5)
\displaystyle \text{Example: }16a^4-b^4=(4a^2+b^2)(4a^2-b^2)=(4a^2+b^2)(2a+b)(2a-b)
\displaystyle \textbf{Tip: }\text{Many expressions should first be rewritten as a difference of squares} \\ \text{before factorising.}
\displaystyle \\

\displaystyle \textbf{Method 5: Sum or Difference of Two Cubes}
\displaystyle \textbf{Identities:}
\displaystyle a^3+b^3=(a+b)(a^2-ab+b^2)
\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)
\displaystyle \textbf{Recognise Common Cubes: }8=2^3,\;27=3^3,\;64=4^3,\;125=5^3
\displaystyle \text{Example: }a^3+27b^3=(a+3b)(a^2-3ab+9b^2)
\displaystyle \text{Example: }8a^3-27b^3=(2a-3b)(4a^2+6ab+9b^2)
\displaystyle \textbf{Application: }\text{These identities are useful in proving divisibility of numbers.}
\displaystyle \\

\displaystyle \textbf{3. Mixed Factorisation}
\displaystyle \text{Many expressions require more than one method of factorisation.}
\displaystyle \text{Common combinations include taking a common factor first, followed by grouping,} \\ \text{difference of squares or cubes.}
\displaystyle \text{Always continue factorising until every factor is irreducible.}
\displaystyle \\

\displaystyle \textbf{4. Standard Identities Used in this Chapter}
\displaystyle a^2-b^2=(a+b)(a-b)
\displaystyle a^3+b^3=(a+b)(a^2-ab+b^2)
\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)
\displaystyle \\

\displaystyle \textbf{5. Strategy for Factorising an Expression}
\displaystyle \text{Step 1: Check for a common factor.}
\displaystyle \text{Step 2: If not, check whether grouping is possible.}
\displaystyle \text{Step 3: If it is a quadratic trinomial, split the middle term.}
\displaystyle \text{Step 4: Check for the difference of two squares.}
\displaystyle \text{Step 5: Check for the sum or difference of two cubes.}
\displaystyle \text{Step 6: If necessary, combine two or more methods.}
\displaystyle \\

\displaystyle \textbf{6. Common Mistakes to Avoid}
\displaystyle \bullet\ \text{Forgetting to take out the H.C.F. first.}
\displaystyle \bullet\ \text{Choosing incorrect numbers while splitting the middle term.}
\displaystyle \bullet\ \text{Applying an identity to an expression that does not match its form.}
\displaystyle \bullet\ \text{Stopping before the expression is completely factorised.}
\displaystyle \bullet\ \text{Missing the second stage of factorisation after using an identity.}
\displaystyle \\

\displaystyle \textbf{7. Learning Outcomes}
\displaystyle \bullet\ \text{Understand the meaning of factorisation and factors.}
\displaystyle \bullet\ \text{Factorise expressions by taking out the common factor.}
\displaystyle \bullet\ \text{Factorise expressions using the grouping method.}
\displaystyle \bullet\ \text{Factorise quadratic trinomials by splitting the middle term.}
\displaystyle \bullet\ \text{Use }b^2-4ac\text{ to determine whether a quadratic is factorisable.}
\displaystyle \bullet\ \text{Apply the identities for the difference of two squares and the sum and difference} \\ \text{of two cubes.}
\displaystyle \bullet\ \text{Factorise complex algebraic expressions by combining suitable methods.}

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