Polynomial sketches, repeated roots and signs
| English | 中文 | Pinyin |
|---|---|---|
| multiplicity/ˌmʌltɪˈplɪsɪti/ | 重数 | chóng shù |
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.
Which description correctly defines multiplicity?
The number of times a root’s linear factor appears in a polynomial.
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.
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.
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.
Find its even-multiplicity root.
The squared factor (x−1)² gives the even-multiplicity root x=1.
Every real root of a polynomial must change the sign of its values.
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.
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.
Evaluate Q(0) for Q(x)=−(x²−1)(x²−4).
Q(0)=−(−1)(−4)=−4.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.