الأسبوع 5 · التحليل: العامل المشترك الأكبر ثم ثلاثية الحدود والمعادلات التربيعية · 3 دروس
1
Factorising by taking out the greatest common factor; solving equations by factorising
Mathematics Prep 2 Term 1, Unit 2, Lessons 1–2
GoalTake out the GCF of numbers and terms, check by expanding, and solve equations like x² = 3x with the zero-product rule.
New words
factorise · write an expression as a product; it is expanding backwardsGCF · the greatest common factor: the biggest expression that divides every termcommon binomial factor · a bracket that appears in every term, such as (b + 5)zero-product rule · if a × b = 0 then a = 0 or b = 0
Explanation
Factorising undoes expanding. To take out the GCF: (1) split each term into factors, (2) find the biggest numbers and letters common to all terms, using the lowest power of each letter, (3) divide each term by it and write GCF × bracket. For 6x² − 15x the GCF is 3x, so 6x² − 15x = 3x(2x − 5). Check by expanding: 3x × 2x − 3x × 5. Take out the whole GCF: 6x − 8y = 2(3x − 4y). A bracket can be a common factor too: a(b + 5) + 2(b + 5) = (b + 5)(a + 2). This also speeds up arithmetic: 19 × 23 + 19 × 27 = 19 × 50 = 950. The zero-product rule solves equations: if (first factor) × (second factor) = 0, one of them is 0. To solve x² = 3x, move everything to one side, x² − 3x = 0, factorise, x(x − 3) = 0, so x = 0 or x = 3. Never divide both sides by x: you would lose x = 0.
Step
For 6x² − 15x
factor each term
2 × 3 × x × x and 3 × 5 × x
common part
3x
divide each term
2x and 5
write
3x(2x − 5)
Worked examples
Factorise 12x³ − 18x². The GCF of 12 and 18 is 6; the lowest power of x is x². So 12x³ − 18x² = 6x²(2x − 3). Check: 6x² × 2x = 12x³ and 6x² × 3 = 18x².
Factorise 3x(x + 2) − 5(x + 2). The bracket (x + 2) is common, so the answer is (x + 2)(3x − 5).
A rectangular plot has width x m and area x² + 5x m². Factorise: x² + 5x = x(x + 5), so its length is x + 5 m. Now solve 2x² + 8x = 0: 2x(x + 4) = 0, so x = 0 or x = −4. Check x = −4: 2 × 16 − 32 = 0.
Class activity
Mental maths. Use common factors to find 73 × 18 + 73 × 82, and 6 × 47 + 6 × 53. Then check each answer by working out the products separately.
Practice
1
Factorise 8x + 12.
4(2x + 3).
2
Factorise 10a²b − 15ab².
5ab(2a − 3b).
3
Find 73 × 18 + 73 × 82 quickly.
73 × (18 + 82) = 73 × 100 = 7300.
4
Solve x² + 5x = 0.
x(x + 5) = 0, so x = 0 or x = −5.
5
Salma solves x² = 4x by dividing by x and gets x = 4. What does she lose?
She loses x = 0. The correct way: x² − 4x = 0, x(x − 4) = 0, so x = 0 or x = 4.
