Equivalent expressions and domain restrictions
| English | 中文 | Pinyin |
|---|---|---|
| factor/ˈfæktə/ | 因式 | yīn shì |
| domain restriction/dəˈmeɪn rɪˈstrɪkʃn/ | 定义域限制 | dìng yì yù xiàn zhì |
A decision before an answer
- Cancelling x-2 from a rational expression simplifies its formula but does not permit the original denominator to be zero.
- Your goal: Factor and expand expressions without changing their values.
Read the relationship
- Expansion distributes each factor across the others; factorisation reverses that process. x²-5x+6=(x-2)(x-3), so the product form exposes zeros while the expanded form exposes coefficients. Choose the form that serves the question rather than treating one representation as always preferable.
- Simplify rational and radical expressions on the original domain.
Simplify (x²-16)/(x-4), with x≠4.
Difference of squares gives (x-4)(x+4).
Use the defining rule
- Cancel factors, never terms in a sum. (x²-4)/(x-2) becomes x+2 only for x≠2, because the original expression is undefined at 2. A restriction may remain after every denominator disappears from the simplified formula. State it before cancelling, then carry it through comparisons or equation solving.
- Use expression structure to identify zeros and coefficients.
For x=-6, √(x²) equals:
The principal root is |x|.
Check the conditions
- Exponent rules depend on their setting: a^m a^n=a^(m+n) and (a^m)^n=a^(mn) in the permitted real domain. Negative exponents form reciprocals and require nonzero bases. With real square roots, √(x²)=|x|, because the principal square root is nonnegative; it is not x for every real x.
- Use expression structure to identify zeros and coefficients.
(x²-9)/(x-3)=(x-3)(x+3)/(x-3)=x+3 for x≠3. At x=3 the original is undefined. For x=-5, √(x²)=5=|x|. In (2x+1)(x-4), the x coefficient is -8+1=-7 and the full expression is 2x²-7x-4.
In (x+2)(x+5), the x coefficient is ____.
The two x terms contribute 5x+2x.
Apply the task format
- To check equivalence, a numerical substitution can expose a mistake, but one matching value does not prove two expressions equal. A valid algebraic identity plus matching domain does. When the question requests a coefficient, expand only as far as needed and account for every contribution to that power.
- Use expression structure to identify zeros and coefficients.
A simplified formula is not automatically an equivalent function on a larger domain; cancelling terms across addition is invalid.
Which answer fits this case?
Factor and expand expressions without changing their values
Matching at x=1 proves two expressions equivalent for all x.
Different expressions can share one value.
Keep the distinctions
- factor 因式 — An expression multiplied by another to form a product.
- domain restriction 定义域限制 — A condition excluding values for which the original expression is undefined.
- Factor and expand expressions without changing their values.
- Simplify rational and radical expressions on the original domain.
- Use expression structure to identify zeros and coefficients.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.