등가 표현 및 정의역 제한 조건 ⟦number⟧
도입됨| English | 의미 |
|---|---|
| factor/ˈfæktə/ | 인수 |
| domain restriction/dəˈmeɪn rɪˈstrɪkʃn/ | 정의역 제한 |
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.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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:
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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 ____.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
값을 변경하지 않고 식을 인수분해 및 전개하기
Matching at x=1 proves two expressions equivalent for all x.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.