Algebraic fractions and excluded values
| English | Français |
|---|---|
| excluded value/eksˈkluːdɪd ˈvæljuː/ | excluded value |
A cancelled fraction can look like a straight line and still have one missing point. 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. Cancelling a factor does not put an undefined input back into the original formula.
- This lesson studies excluded value 排除值: An input for which an original denominator or divisor is zero.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines excluded value?
An input for which an original denominator or divisor is zero.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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).
5 is allowed; after factor cancellation, F(5)=5+3=8.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Cancelling a denominator factor always makes its excluded input valid. This claim is false. Explain which definition or assumption it violates.
Find 1/(x-1)+2/(x+1) at x=3.
1/2+2/4=1, agreeing with(3x-1)/(x²-1).
Cancelling a denominator factor always makes its excluded input valid.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many allowed solutions does (x²-9)/(x-3)=6 have?
The only reduced-equation candidate is3; the original denominator is zero there.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An input for which an original denominator or divisor is zero. Choose the relationship, show the method, check its assumptions and interpret the result.