Learning goal and prerequisites
Infer crossings, sign intervals and end behavior from a factored polynomial. Before starting, review polynomial structure.
Method and conditions
Locate distinct real zeros, label each multiplicity, and test one interval. Odd multiplicities switch sign; even multiplicities preserve it. Use the leading term only for far-left and far-right behavior.
Worked example: show the decisions
For P(x)=−2(x+1)²(x−3)³, degree 5 and leading coefficient −2 imply positive values far left and negative values far right. The graph touches at −1 but crosses at 3. On (−1,3), the squared factor is positive and the cubic factor negative, so P is positive.
Contrast: why a plausible shortcut fails
The number of distinct zeros is two; the total multiplicity is five; the number of crossings is one. These are three different questions. A complex factor x²+4 contributes degree two but no real zero.
Transfer: try before revealing the answer
For Q(x)=(x−2)²(x+4), identify crossings and count integers satisfying Q(x)<0 between −6 and 4 inclusive.
Transfer solution and check
Only −4 is a crossing. Q is negative for x<−4, giving integers −6 and −5: two values. x=2 is a zero, not negative.
How many integers x satisfy (x−3)(x−9)<0?
- The product is negative strictly between the roots; count the five interior integers.
- Using these quantities: 6-1 = 5.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Compare the other answer choices
Count the roots rather than the integers giving negative products.
Use interval length rather than the number of interior integers.
Include both zero endpoints in the strict inequality.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance