Chapter 7

Algebra: Modulus, Polynomials and Partial Fractions (Pure 3)

Modulus equations and inequalities, polynomial division with the factor and remainder theorems, partial fractions in all three of their forms, and what to do when an equation cannot be solved exactly.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)การหารพหุนามและทฤษฎีบทเศษเหลือตรงกับหลักสูตรไทย ม.5 · เศษส่วนย่อยและวิธีเชิงตัวเลขไม่มีในหลักสูตรไทยระดับมัธยม
Common Core (US): A-APR, A-REI
Cambridge: 9709 Pure Mathematics 3 — Algebra; Numerical solution of equations

7.1 Modulus equations and inequalities

The two moves

∣ a ∣ = ∣ b ∣ ⟺ a^{2} = b^{2} ∣ x - a ∣ < b ⟺ a - b < x < a + b

The second reads as " $x$ is within $b$ of $a$ ", which is worth saying aloud — it turns most modulus inequalities into a sentence about distance.

ตัวอย่าง

Solve $∣ 2 x - 1 ∣ > ∣ x + 4 ∣$ .

วิธีทำ

1.Square both sides: $4 x^{2} - 4 x + 1 > x^{2} + 8 x + 16$
2. $3 x^{2} - 12 x - 15 > 0$
3. $x^{2} - 4 x - 5 > 0$ , so $(x - 5) (x + 1) > 0$
4.A positive parabola is above the axis outside its roots

ตอบ $x < - 1$ or $x > 5$

ข้อควรระวัง

Squaring preserves the inequality only because both sides are moduli and therefore non-negative. Squaring $∣ x ∣ > x - 3$ would not be safe, and every answer would have to be checked in the original.

📖 3 more sections of this chapter are being checked

The whole chapter is written; we read it back section by section before releasing it, because maths nobody has checked should not be on a public page. What is open is complete — nothing here is abridged.

Read it — now practise it

Generate a worksheet on this topic, or sit a timed mock paper.

Build a worksheet Practice arena