MathematicsChapter 5: Algebraic Fractions
Long Division (named Skill)
Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom.
Learning objectives
- Explain long division (named skill) using the correct definitions, conditions, and notation.
- Apply divide leading terms, remainder 0 => factor correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom. x4 - 10x2 + 9x2 - 2x - 3 = x2 + 2x - 3 in the MCQ list. (x-y)3x-y = (x-y)^2. Remainder 0 means the bottom was a factor.
Formula
y = (x - y)2
Where / symbols
- LONG DIVISION (NAMED SKILL)
- Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom. (x^4 - 10x^2 + 9)/(x^2 - 2x - 3) = x^2 + 2x - 3 in the MCQ list. (x-y)^3 / (x-y) = (x-y)^2. Remainder 0 means the bottom was a factor.
- QUOTIENT
- Leading x^4 / x^2 = x^2. Multiply back, subtract, repeat. If the paper allows factoring both, that is faster and should match.
- REMAINDER
- If remainder is not 0, write quotient + remainder/divisor, like numbers. Many Grade 10 examples divide evenly.
Relationship
- remainder 0 => factor
Overview
Leading term, then factor check:
Quotient
Leading x^4x2 = x^2. Multiply back, subtract, repeat. If the paper allows factoring both, that is faster and should match.
Examples x^2+2x-3 · (x-y)^3/(x-y)=(x-y)^2 · Not only the leading term as the whole answer
Remainder
If remainder is not 0, write quotient + remainder/divisor, like numbers. Many Grade 10 examples divide evenly.
Examples Factor first if you see it · Check: divisor * quotient = dividend · x!= roots of divisor
WHAT — LONG DIVISION (NAMED SKILL)
Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom. x4 - 10x2 + 9x2 - 2x - 3 = x2 + 2x - 3 in the MCQ list. (x-y)3x-y = (x-y)^2. Remainder 0 means the bottom was a factor. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Long Division (Named Skill) matters because rational expressions combine algebra with fraction rules. Factoring, restrictions, and checking are essential because an apparently simple answer may still exclude some values.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through QUOTIENT, then REMAINDER. Worked: (x - y)^3x - y. (x - y)3 = (x - y)(x - y)2 cancel (x - y) with x != y = (x - y)2 = x^2 - 2xy + y^2 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- (x - y)frac3x - y.
- (x - y)3 = (x - y)(x - y)2
- cancel (x - y) with x != y
- = (x - y)2
- = x^2 - 2xy + y^2
In simple words
Division by x-y is undefined at x=y even if the cubic is 0 there. The simplified polynomial extends the values; the original fraction does not.
Key points
- divide leading terms
- remainder 0 => factor
- leading over leading
- R=0 nice
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Long Division (named Skill)
Sticky Notes · Quick Revision
Core Idea
- Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom. x4 - 10x2 + 9x2 - 2x - 3 = x2 + 2x - 3 in the MCQ list. (x-y)3x-y = (x-y)^2. Remainder 0 means the bottom was a factor.
QUOTIENT
- Leading x^4x2 = x^2. Multiply back, subtract, repeat. If the paper allows factoring both, that is faster and should match.
REMAINDER
- If remainder is not 0, write quotient + remainder/divisor, like numbers. Many Grade 10 examples divide evenly.
Worked Method
- Worked: (x - y)frac3x - y.
- (x - y)3 = (x - y)(x - y)2
- cancel (x - y) with x != y
- = (x - y)2
- = x^2 - 2xy + y^2
Common Misconception
- Division by x-y is undefined at x=y even if the cubic is 0 there. The simplified polynomial extends the values; the original fraction does not.
FBISE Exam Check
- Division by x-y is undefined at x=y even if the cubic is 0 there. The simplified polynomial extends the values; the original fraction does not.
60-second revision
- Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom.
- QUOTIENT: Leading xfrac4x2 = x2.
- REMAINDER: If remainder is not 0, write quotient + remainder/divisor, like numbers.
- Worked-method reminder: = x^2 - 2xy + y^2
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