Algebra: Expressions and Formulae

Expanding, factorising in every form the syllabus asks for, algebraic fractions, and rearranging a formula when the letter you want appears twice.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ครอบคลุม ม.2 บทที่ 6 และ 11 · ม.3 บทที่ 2 การแยกตัวประกอบดีกรีสูง
Common Core (US): A-SSE, A-APR
Cambridge: IGCSE 0580 — Topic 2: Algebra and graphs (Core and Extended)

2.1 Expanding and factorising

The order to try factorising in

1Common factor — always first, and often the only step needed
2Two terms: difference of two squares, $a^{2} - b^{2} = (a + b) (a - b)$
3Three terms: find the pair multiplying to $a c$ and adding to $b$
4Four terms: group in pairs and look for a shared bracket
5Multiply back out to check

ตัวอย่าง 1

Factorise $6 x^{2} - x - 2$

วิธีทำ

1. $a c = 6 \times (- 2) = - 12$ ; need two numbers multiplying to $- 12$ , adding to $- 1$ : $3$ and $- 4$
2.Split: $6 x^{2} + 3 x - 4 x - 2$
3.Group: $3 x (2 x + 1) - 2 (2 x + 1)$

ตอบ $(2 x + 1) (3 x - 2)$

ตัวอย่าง 2

Factorise $a x + a y - b x - b y$

วิธีทำ

1.Group in pairs: $a (x + y) - b (x + y)$
2.The bracket $(x + y)$ is common

ตอบ $(x + y) (a - b)$

ข้อควรระวัง

If grouping leaves two brackets that differ, the sign was taken out wrongly on the second pair. In the example above it has to be $- 2$ , not $2$ , or the brackets will not match.

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