MathematicsChapter 5: Algebraic Fractions
Factoring For Fractions
You cancel common factors, not common terms.
Learning objectives
- Explain factoring for fractions using the correct definitions, conditions, and notation.
- Apply factor then cancel, not terms correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
You cancel common factors, not common terms. So factor first: difference of squares, trinomials, difference of cubes x^3-y^3=(x-y)(x^2+xy+y^2). Canceling x in x2+xx3+x needs x(x+1) over x(x^2+1).
Mathematical form
x = x(x + 1) bottom
x = x(x2 + 1) x(x + 1)x(x2 + 1) cance
Where / symbols
- FACTORING FOR FRACTIONS
- You cancel common factors, not common terms. So factor first: difference of squares, trinomials, difference of cubes x^3-y^3=(x-y)(x^2+xy+y^2). Canceling x in (x^2+x)/(x^3+x) needs x(x+1) over x(x^2+1).
- SQUARES AND TRINOMIALS
- x^2-49=(x-7)(x+7). x^2-x-20=(x-5)(x+4). 2x^2+7x+3=(2x+1)(x+3). 2x^2-11x-21=(2x+3)(x-7).
- CUBES AND COMMON x
- x^3-y^3=(x-y)(x^2+xy+y^2). Pull x from every term if it is common. Cancel the factor, and note x!=0 if you cancelled x.
Overview
Patterns you need:
Squares and Trinomials
x2-49=(x-7)(x+7). x2-x-20=(x-5)(x+4). 2x2+7x+3=(2x+1)(x+3). 2x2-11x-21=(2x+3)(x-7).
Examples x^2-49 · x^2-x-20 · (2x+3)(x-7)
CUBES AND COMMON x
x^3-y^3=(x-y)(x^2+xy+y^2). Pull x from every term if it is common. Cancel the factor, and note x!=0 if you cancelled x.
Examples x^3-y^3 · cancel x, x!=0 · Not cancel x in x+1 over x+2
WHAT — FACTORING FOR FRACTIONS
You cancel common factors, not common terms. So factor first: difference of squares, trinomials, difference of cubes x^3-y^3=(x-y)(x^2+xy+y^2). Canceling x in x2+xx3+x needs x(x+1) over x(x^2+1). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Factoring For Fractions 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 SQUARES AND TRINOMIALS, then CUBES AND COMMON x. Worked: reduce x2 + xx3 + x. top: x2 + x = x(x + 1) bottom: x3 + x = x(x2 + 1) x(x + 1)x(x2 + 1) cancel x (with x != 0) = x + 1x2 + 1 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- reduce x2 + xx3 + x.
- top: x2 + x = x(x + 1)
- bottom: x3 + x = x(x2 + 1)
- x(x + 1)x(x2 + 1)
- cancel x (with x != 0)
- = x + 1x2 + 1
In simple words
You may cancel factors only. Canceling the x in x+1x as if it were 1+1 is wrong.
Key points
- factor then cancel
- not terms
- two numbers / grouping
- cubes formula
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Factoring For Fractions
Sticky Notes · Quick Revision
Core Idea
- You cancel common factors, not common terms. So factor first: difference of squares, trinomials, difference of cubes x^3-y^3=(x-y)(x^2+xy+y^2). Canceling x in x2+xx3+x needs x(x+1) over x(x^2+1).
SQUARES AND TRINOMIALS
- x2-49=(x-7)(x+7). x2-x-20=(x-5)(x+4). 2x2+7x+3=(2x+1)(x+3). 2x2-11x-21=(2x+3)(x-7).
CUBES AND COMMON x
- x^3-y^3=(x-y)(x^2+xy+y^2). Pull x from every term if it is common. Cancel the factor, and note x!=0 if you cancelled x.
Worked Method
- Worked: reduce x2 + xx3 + x.
- top: x2 + x = x(x + 1)
- bottom: x3 + x = x(x2 + 1)
- x(x + 1)x(x2 + 1)
- cancel x (with x != 0)
- = x + 1x2 + 1
Common Misconception
- You may cancel factors only. Canceling the x in x+1x as if it were 1+1 is wrong.
FBISE Exam Check
- You may cancel factors only. Canceling the x in x+1x as if it were 1+1 is wrong.
60-second revision
- You cancel common factors, not common terms.
- SQUARES AND TRINOMIALS: x2-49=(x-7)(x+7).
- CUBES AND COMMON x: x3-y3=(x-y)(x2+xy+y2).
- Worked-method reminder: = x + 1x2 + 1
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