SAT-41: Simplifying Rational Expressions & Polynomial Division

SAT-41: Simplifying Rational Expressions & Polynomial Division

Description: A rational expression is a fraction whose top and bottom are polynomials. You simplify it the same way you simplify number fractions: factor, then cancel common factors. When it won't cancel cleanly, use polynomial long division.

Method 1 — Factor and cancel

1. Factor the numerator and denominator fully (review SAT-31, SAT-32).
2. Cancel any factor that appears on both top and bottom.
3. Note the restricted values (where the original denominator was 0).

(Oʻzbekcha: suratni va maxrajni koʻpaytuvchilarga ajrating, soʻng umumiy koʻpaytuvchilarni qisqartiring.)

Worked Example (factor and cancel)

Simplify (x² − 9) / (x² + 5x + 6).

• Factor top: x² − 9 = (x − 3)(x + 3) (difference of squares).
• Factor bottom: x² + 5x + 6 = (x + 2)(x + 3).
• Cancel (x + 3): result = (x − 3) / (x + 2), with x ≠ −2, −3.

(Oʻzbekcha: faqat koʻpaytuvchilarni qisqartiring — hadlarni emas.)

Method 2 — Polynomial long division

When factoring/cancelling isn't possible, divide like you divide numbers.

Divide (x² + 5x + 6) ÷ (x + 2):

• x² ÷ x = x. Multiply: x(x + 2) = x² + 2x. Subtract → 3x + 6.
• 3x ÷ x = 3. Multiply: 3(x + 2) = 3x + 6. Subtract → 0.
• Quotient: x + 3, remainder 0.

Tip: a remainder of 0 means the divisor is a factor of the polynomial — this idea leads into the Factor Theorem (SAT-42). (Oʻzbekcha: qoldiq 0 boʻlsa, boʻluvchi — koʻpaytuvchi.)

Practice

Simplify (x² − 4) / (x − 2).

Key words — Kalit soʻzlar

• Rational expression — ratsional ifoda
• Polynomial — koʻphad
• Numerator — surat
• Denominator — maxraj
• Factor — koʻpaytuvchi / koʻpaytuvchilarga ajratish
• Cancel — qisqartirish
• Difference of squares — kvadratlar ayirmasi
• Long division — burchak (uzun) boʻlish
• Quotient — boʻlinma
• Remainder — qoldiq

Summary

• Simplify a rational expression by factoring top and bottom, then cancelling common factors.
• Only cancel factors, never individual terms.
• When it won't cancel, use polynomial long division (quotient + remainder).
• A remainder of 0 means the divisor is a factor.