Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 01) Algebraic Tools cover? A: Core expansion, factorisation, and simplification identities that power the rest of IP Elementary Mathematics.
Keep algebra tidy so every later topic - graphs, trigonometry, variation - stays manageable. These identities should come out automatically.
Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.
Key skills to lock in
Apply the distributive law: a(b+c)=ab+ac.
Recognise perfect-square and difference-of-squares patterns, e.g. x2−9=(x−3)(x+3)
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Factor by grouping to reveal hidden common factors.
Simplify expressions with multiple parentheses before substituting values.
Worked example - Rewrite for evaluation
Rewrite 3(2x−5)+4(x+1) in simplified form, then find its value when x=−2.
Expand each product: 3(2x−5)=6x−15 and 4(x+1)=4x+4.
Combine like terms: (6x−15)+(4x+4)=10x−11.
Substitute x=−2: 10(−2)−11=−20−11=−31.
So the simplified expression is 10x−11, and its value at x=−2 is −31.
Worked example - Spot a perfect square / difference of squares
Simplify 3y+59y2−25 and evaluate it at y=−1.
Recognise 9y2−25 as a difference of squares: 9y2=(3y)2 and 25=52.
Factor the numerator: 9y2−25=(3y−5)(3y+5).
Cancel the common factor (3y+5) with the denominator, noting y=−35: we get 3y−5
Substitute y=−1: 3(−1)−5=−3−5=−8.
The simplified expression is 3y−5, and it evaluates to −8 when y=−1.
Worked example - Factor by grouping to solve an equation
Solve 4x(3x−2)−5(3x−2)=0.
Factor by grouping: both terms share (3x−2), so write (3x−2)(4x−5)=0.
Apply the zero-product property: either 3x−2=0 or 4x−5=0.
Solve each linear equation:
3x−2=0⇒x=32.
4x−5=0⇒x=45
Therefore, x=32 or x=45.
Practice Quiz
Keep your factorisation, expansion, and zero-product property instincts sharp with auto-marked drills.
Try this
Factorise 2x2−5x−12 completely and state the values of x that make the expression zero.
IP EMaths Notes (Upper Sec, Year 3-4): 01) Algebraic Tools
