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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.
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Long Division (named Skill)
What is it?
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.
In simple words
- 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
Important terms
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.
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.
What you should learn
- 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.
Examples
- x^2+2x-3
- (x-y)frac3x-y=(x-y)2
- Not only the leading term as the whole answer
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.
Recommended for Class 10
Full school-level explanation
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.
Leading term, then factor check:
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.
QUOTIENT
Leading x^4x2 = x^2. Multiply back, subtract, repeat. If the paper allows factoring both, that is faster and should match.
Remember
- leading over leading
Examples
- x^2+2x-3
- (x-y)frac3x-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.
Remember
- R=0 nice
Examples
- Factor first if you see it
- Check: divisor * quotient = dividend
- x!= roots of divisor
Remember
- divide leading terms
- remainder 0 => factor
- x^2 - 2xy + y^2
- x != y
- leading over leading
- R=0 nice
- divide leading terms
- remainder 0 => factor
How the ideas connect
Worked: (x - y)frac3x - y. (x - y)3 = (x - y)(x - y)2 cancel (x - y) with x != y = (x - y)2 = x2 - 2xy + y2
Deeper understanding
Go beyond memorising Long Division (named Skill). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: (x - y)frac3x - y. (x - y)3 = (x - y)(x - y)2 cancel (x - y) with x != y = (x - y)2 = x2 - 2xy + y2
Common mistakes
Exam focus
Core Idea
QUOTIENT
REMAINDER
Worked Method
Exam focus
Quick revision
- divide leading terms
- remainder 0 => factor
- x^2 - 2xy + y^2
- x != y
- LONG DIVISION (NAMED SKILL): Polynomial long division (or factoring) divides a high-degree top by a lower-degree bottom. (x^4 - 1
- QUOTIENT: Leading x^4x2 = x^2. Multiply back, subtract, repeat. If the paper allows factoring both, that i
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.
For long answers on Long Division (named Skill): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- divide leading terms
- remainder 0 => factor
- x^2 - 2xy + y^2
- x != y
Important formulas
- divide leading terms
- remainder 0 => factor
- x^2 - 2xy + y^2
Common exam traps
- 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.
- Not only the leading term as the whole answer
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
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.
- Mistake
Not only the leading term as the whole answer
- Mistake
Do not treat QUOTIENT and REMAINDER as the same step; each has a different purpose.
- Mistake
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.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
divide leading terms
remainder 0 => factor
x^2 - 2xy + y^2
x != y
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-
remainder 0 => factor
- 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-
- Formula
- remainder 0 => factor
- Example
- x^2+2x-3
- Critical warning
- Division by x-y is undefined at x=y even if the cubic is 0 there. The simplified polynomial extends the values; the orig
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Exam Practice
Define
Q1. Define Long Division (named Skill).
Explain
Q2. Explain: QUOTIENT.
Example
Q3. Give one clear example of Long Division (named Skill) and say why it fits.
Mistake
Q4. State one common mistake students make about Long Division (named Skill).
Exam note
Q5. Write a short exam note on Long Division (named Skill): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.