多項式の概略図、重複根および符号
導入済み| English | 意味 |
|---|---|
| multiplicity/ˌmʌltɪˈplɪsɪti/ | 重数 |
Two roots can appear in a factorised formula but make very different marks on its graph. Which one crosses the axis?
- Two roots can appear in a factorised formula but make very different marks on its graph. Which one crosses the axis?
- This lesson studies multiplicity 重数: The number of times a root’s linear factor appears in a polynomial.
Choose the mathematical structure
- Find real roots and their multiplicities, the y-intercept and the leading term. A root of odd multiplicity changes the sign and crosses the x-axis; a root of even multiplicity keeps the sign and touches the axis. For a positive leading coefficient, an odd-degree curve goes down on the far left and up on the far right; an even-degree curve goes up at both ends. A negative leading coefficient reverses these directions.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
P(x)=(x+2)(x-1)²において、なぜ曲線はx=1で接して crossing しないのか？
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
x=1付近では、因子 (x-1)² は両側で非負であるため、P の符号は変化せず、曲線は接して折り返す。単根 -2 は奇数次多重性を持つため、軸を横切る。
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 P(x)=(x+2)(x−1)², roots are −2 with multiplicity 1 and 1 with multiplicity 2. The y-intercept is P(0)=2. The leading term is x³, so the left tail falls and the right rises. P crosses at −2 and touches at 1. It is negative for x<−2, positive for −2<x<1 and x>1, and zero at both roots. For Q(x)=−(x²−1)(x²−4), the four simple roots are −2,−1,1,2, the y-intercept is −4 and the leading term is −x⁴. Both tails fall. Q is positive on (−2,−1) and (1,2), and negative on the other three root-separated intervals.
多項式の概略図、重複根および符号
実の根及其びその重数、y切片、および最高次項を求めよ。
因数分解された式をそのグラフまたは展開係数に結びつける。
P(x)=(x+2)(x−1)² の y切片を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
P(0)=(0+2)(0−1)²=2.
Test a tempting shortcut
- The number of distinct roots need not equal the degree. A repeated root is one x-intercept even though its factor occurs more than once. Do not alternate the sign at an even-multiplicity root. A sketch shows structure; exact turning-point coordinates require further analysis, not guessed readings from the picture.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every real root of a polynomial must change the sign of its values. This claim is false. Explain which definition or assumption it violates.
偶数重数の根を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
二乗因子 (x−1)² は、偶数重数の根 x=1 を与える。
多項式のすべての実の根は、その値の符号を変えなければならない。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
相異なる根の個数が次数と等しいとは限らない。重複根とは、因子が複数回現れていても、x軸との交点が一つであることを指す。偶数重数の根で符号を交互に変えてはならない。概略図は構造を示すが、正確な極点座標を得るには further analysis が必要であり、図からの推定読み取りに依存してはならない。
Interpret a new situation
- Graphical solutions of P(x)=k are intersections of the curve with y=k. For k=0 the factored roots are exact; for another k a graph may only estimate solutions. Use the sign intervals to solve P(x)≤0: x≤−2 or x=1. That isolated root matters even though its neighbouring values are positive.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Q(0) を計算せよ。ただし Q(x)=−(x²−1)(x²−4) である。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Q(0)=−(−1)(−4)=−4.
各スケッチの特徴に対応する要因を結びつけよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
多重性が横切りか接かのどちらかを制御し、首位項が両端の方向を決定する。正確な極点の位置は推測ではなく微分積分を用いる必要がある。
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.
The number of times a root’s linear factor appears in a polynomial. Choose the relationship, show the method, check its assumptions and interpret the result.