Polynomial sketches, repeated roots and signs
Введено| English | Значение |
|---|---|
| multiplicity/ˌmʌltɪˈplɪsɪti/ | кратность корня |
Which one crosses the axis?
- Two roots can appear in a factorised formula but make very different marks on its graph.
- This lesson studies multiplicity 重数: The number of times a root’s linear factor appears in a polynomial.
The method
- 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.
For P(x)=(x+2)(x-1)², why does the curve touch but not cross at x=1?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Near x=1 the factor (x-1)² is nonnegative on both sides, so the sign of P is unchanged - the curve touches and turns. The simple root -2, with odd multiplicity, crosses.
A worked case
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.
Polynomial sketches, repeated roots and signs
Find real roots and their multiplicities, the y-intercept and the leading term
Connect a factored expression to its graph or expansion coefficients.
Find the y-intercept of P(x)=(x+2)(x−1)².
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
P(0)=(0+2)(0−1)²=2.
Where it goes wrong
- 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.
Not true: Every real root of a polynomial must change the sign of its values. A repeated root of even multiplicity touches the axis without changing sign, as y=x² does at 0.
Find its even-multiplicity root.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The squared factor (x−1)² gives the even-multiplicity root x=1.
A repeated root of even multiplicity touches the axis without changing sign, as y=x² does at 0.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
True. A repeated root of even multiplicity touches the axis without changing sign, as y=x² does at 0.
Applying it
- 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.
Evaluate Q(0) for Q(x)=−(x²−1)(x²−4).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Q(0)=−(−1)(−4)=−4.
Match each sketch feature to its source.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Multiplicity controls crossing versus touching, and the leading term fixes both tails; exact turning points need calculus, not guesses from a sketch.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Multiplicity: The number of times a root’s linear factor appears in a polynomial.
- Find real roots and their multiplicities, the y-intercept and the leading term.
- Watch for: the number of distinct roots need not equal the degree.