線形、二次式および多項式の関係
導入済み| English | 意味 |
|---|---|
| sign interval/saɪn ˈɪntəvl/ | 符号区間 |
| polynomial degree/ˌpɒlɪˈnəʊmɪəl dɪˈɡriː/ | 多項式の次数 |
A decision before an answer
- The roots of a quadratic mark boundaries. They do not by themselves tell you which interval makes the expression negative.
- Your goal: Choose equivalent expanded, factored or vertex forms for a task.
Read the relationship
- Translate relationships into expressions before manipulating them. A fixed fee plus a per-unit charge is an affine linear model, while a product of dimensions can create a quadratic. An equivalent form should preserve the original expression on its domain. Factor common terms and recognise identities before expanding every product.
- Solve linear and polynomial equations using structure.
x(x-5)=0 has solutions:
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Either factor may be zero.
Use the defining rule
- A linear equation can have one, no or infinitely many solutions after simplifying; check zero coefficients before dividing. Multiplying or dividing an inequality by a negative reverses its direction. A polynomial product equals zero when at least one factor is zero. Keep all factors, including an extracted x, to avoid losing a zero root.
- Determine quadratic inequality regions from signs rather than roots alone.
(x-1)(x-4)<0 holds on:
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The factors have opposite signs strictly between the roots.
Check the conditions
- A quadratic’s factored form exposes roots and its vertex form exposes an extremum. For an inequality, order the real roots and test the sign in each interval. Include boundaries only for ≤ or ≥. A parabola opening upward is negative between two distinct real roots and positive outside them.
- Determine quadratic inequality regions from signs rather than roots alone.
x²-5x+6=(x-2)(x-3), so x²-5x+6<0 gives 2<x<3. x³-4x=x(x-2)(x+2), giving roots -2,0,2. The expression 2(x-3)²+5 has minimum 5 at x=3. In (2x+1)(x-4), the x coefficient is -8+1=-7.
The minimum of (x+2)²+7 is ____.
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The squared term is nonnegative and can equal zero.
Apply the task format
- Use substitution into the original relation to check candidate values. An expanded coefficient question may require combining several contributions to the same power. A degree-three polynomial need not require a general cubic formula: a common factor or known root can reduce it to simpler factors. Choose the method based on structure.
- Determine quadratic inequality regions from signs rather than roots alone.
Do not omit an extracted factor, include a strict-inequality boundary, or report a vertex x-coordinate when the question asks for the minimum value.
このケースに当てはまる答えはどれか？
タスクに応じて等価な展開形、因数分解形または頂点形のいずれかを選択する。
Dividing an inequality by -2 leaves its direction unchanged.
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A negative multiplier reverses order.
Keep the distinctions
- sign interval 符号区间 — A region between roots on which an expression keeps one sign.
- polynomial degree 多项式次数 — The highest variable power with a nonzero coefficient after simplification.
- Choose equivalent expanded, factored or vertex forms for a task.
- Solve linear and polynomial equations using structure.
- Determine quadratic inequality regions from signs rather than roots alone.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.