SAT-42: The Remainder Theorem and the Factor Theorem

SAT-42: The Remainder Theorem and the Factor Theorem

Description: In SAT-41 you divided polynomials the long way. These two theorems let you get the same information by just plugging in a number — no long division needed.

The Remainder Theorem

If you divide a polynomial P(x) by (x − a), the remainder is exactly P(a).

So to find a remainder, you don't divide — you evaluate. (Oʻzbekcha: P(x) ni (x − a) ga boʻlgandagi qoldiq — bu P(a) qiymati.)

The Factor Theorem

(x − a) is a factor of P(x) if and only if P(a) = 0.

This is just the special case where the remainder is 0: no remainder means it divides evenly. (Oʻzbekcha: agar P(a) = 0 boʻlsa, (x − a) — P(x) ning koʻpaytuvchisi.)

Worked Example (remainder)

Find the remainder when P(x) = x³ − 2x² + 5 is divided by (x − 2).

• Here a = 2. Compute P(2) = 2³ − 2(2)² + 5 = 8 − 8 + 5 = 5.
• The remainder is 5 (no division required).

Worked Example (factor)

Is (x − 1) a factor of P(x) = x³ + 2x − 3?

• Compute P(1) = 1 + 2 − 3 = 0. Since P(1) = 0, yes, (x − 1) is a factor.

(Oʻzbekcha: P(1) = 0 boʻlgani uchun (x − 1) koʻpaytuvchi ekan.)

Practice

What is the remainder when P(x) = 2x² + 3x − 4 is divided by (x + 1)?

Key words — Kalit soʻzlar

• Polynomial — koʻphad
• Remainder Theorem — qoldiq haqidagi teorema
• Factor Theorem — koʻpaytuvchi haqidagi teorema
• Remainder — qoldiq
• Factor — koʻpaytuvchi
• Evaluate — qiymatini hisoblash
• Divide evenly — qoldiqsiz boʻlinish
• Root — ildiz
• If and only if — faqat va faqat shu holda

Summary

• Remainder Theorem: remainder of P(x) ÷ (x − a) is P(a).
• Factor Theorem: (x − a) is a factor ⇔ P(a) = 0.
• Watch the sign: dividing by (x + 1) means a = −1.