Algebraic fractions and excluded values
Введено| English | Значение |
|---|---|
| excluded value/eksˈkluːdɪd ˈvæljuː/ | исключённое значение |
Cancelling a factor does not put an undefined input back into the original formula.
- A cancelled fraction can look like a straight line and still have one missing point.
- This lesson studies excluded value 排除值: An input for which an original denominator or divisor is zero.
The method
- Factor every numerator and denominator before cancelling a common factor.
- A term joined by addition is not a cancellable factor.
- Record exclusions from the original expression first.
- For addition or subtraction use a common denominator, retaining brackets around the entire numerator.
- Multiply factored numerators and denominators.
- To divide, multiply by the reciprocal and exclude inputs making the divisor zero as well as inputs making any original fraction undefined.
For F(x)=(x²-9)/(x-3), why is F(3) undefined even though the simplified form x+3 gives 6?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
F(x)=x+3 holds only where the original expression exists: x≠3. At x=3 the original divides by zero, so 6 is the limit-like value of the simplified form, not F(3). Exclusions must be recorded before cancelling.
A worked case
For F(x)=(x²-9)/(x-3), factor x²-9=(x-3)(x+3). Thus F(x)=x+3 for x≠3; F(5)=8, but F(3) is undefined, not 6. For addition, 1/(x-1)+2/(x+1)=[(x+1)+2(x-1)]/[(x-1)(x+1)]=(3x-1)/(x²-1), with x≠±1. At x=3, both forms give1. For subtraction, the numerator is (x+1)-2(x-1)=3-x, so the minus sign acts on both terms. Multiplication [(x-1)/(x+2)]×[(x+2)/(x+1)] gives(x-1)/(x+1), but the original excludes x=-2 and x=-1. Division [(x²-1)/(x²+3x+2)]÷[(x-1)/(x+2)] gives1, yet x=-2,-1,1 are excluded: x=1 makes the divisor zero. Finally F(x)=6 reduces to x+3=6. Its only candidate x=3 is forbidden, so this equation has no solution. Check candidates in the original equation before reporting them.
Algebraic fractions and excluded values
Factor every numerator and denominator before cancelling a common factor
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
For F(x)=(x²-9)/(x-3), find F(5).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
5 is allowed; after factor cancellation, F(5)=5+3=8.
Where it goes wrong
- Never cancel the x in (x+2)/x.
- A simplified denominator does not show all original restrictions.
- In a division task, check when the divisor is zero.
- Cross-multiplication can produce an excluded candidate; a formal root is not automatically a solution.
Not true: Cancelling a denominator factor makes its excluded input valid. An input that makes the original denominator zero stays excluded, even after cancelling.
Find 1/(x-1)+2/(x+1) at x=3.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
1/2+2/4=1, agreeing with(3x-1)/(x²-1).
Cancelling a denominator factor makes its excluded input valid.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
An input that makes the original denominator zero stays excluded, even after cancelling.
Applying it
- 7357 B6 covers rational-expression manipulation, cancellation and algebraic division by linear expressions.
- State the original restrictions, including zero-divisor exclusions, and check every equation candidate in the original domain.
- Combine this lesson with polynomial division for an improper rational expression.
How many allowed solutions does (x²-9)/(x-3)=6 have?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The only reduced-equation candidate is3; the original denominator is zero there.
Match each algebraic-fraction operation to its rule.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Exclusions come from every original denominator and every divisor; a simplified result keeps them all.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Excluded value: An input for which an original denominator or divisor is zero.
- Factor every numerator and denominator before cancelling a common factor.
- Watch for: never cancel the x in (x+2)/x.