Use factors and leading terms to predict a graph without assuming its entire shape.
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Consider P(x)=(x+2)(x−1)². Find its zeros without expanding. The factor x+2 vanishes at −2; the squared factor vanishes at 1. There are two distinct real roots, although the polynomial has degree three. Explain where the extra degree appears.
The root 1 has multiplicity two because its linear factor occurs twice. Multiplicity is information about the factorization, not an instruction to enter the same solution twice in a set.
Predict
Predict whether the graph changes sign at both roots. A squared real quantity is positive on either side of its zero. A first power changes sign as its input passes through zero. Near each root, inspect the other factors too; their signs determine whether the graph is above or below the axis.
Experiment
Construct a sign chart using one test input in each interval separated by the roots.
Interval
Test x
Sign of x+2
Sign of (x−1)²
Sign of P(x)
Below −2
−3
Negative
Positive
Negative
Between −2 and 1
0
Positive
Positive
Positive
Above 1
2
Positive
Positive
Positive
Draw a rough graph crossing at −2 and touching without crossing at 1. Mark P(0)=2. Do not invent exact turning-point coordinates from this chart: sign and intercept information constrain the shape without determining every detail of a sketch.
Change the squared factor to a cube. Predict the changed sign behavior at 1 and the new degree before recalculating. An odd power changes sign; an even power does not.
Understand
Near a real root r of a nonzero polynomial, write P(x)=(x−r)^m H(x), where m is a positive integer and H(r) is nonzero. In a sufficiently small neighborhood, H keeps the same sign. Thus odd m causes a sign change across r, while even m does not. This explains crossing behavior without assuming a particular graphing window.
The degree is the highest exponent with nonzero coefficient, and the leading coefficient multiplies that power. At large input magnitudes the leading term controls polynomial end behavior. A positive even-degree leading term sends both ends upward. A negative even-degree leading term sends both downward. Odd degree gives opposite ends, with the leading coefficient choosing their orientation.
For −3(x−2)²(x+1)³, degrees add to five and the leading coefficient is −3. The left end rises and the right end falls. The root 2 has even multiplicity, while −1 has odd multiplicity. These facts can all be read without expanding the entire expression.
Over the complex numbers, a degree-n polynomial has n roots counted with multiplicity. This does not promise n distinct real roots. For real coefficients, nonreal roots occur in conjugate pairs. The polynomial (x−2)(x²+1) has roots 2, i and −i; only 2 appears as a real graph intercept. The zero polynomial is an exception to finite root counting: it vanishes everywhere and has no ordinary finite degree.
Apply
In practice, report distinct roots as a set and multiplicities separately in your reasoning. Use a sign chart to justify crossings. The complex-root task asks you to verify a factorization and explain why a real graph cannot display every complex solution.
Check a rough sketch against its intercept, signs and both ends. A picture that satisfies only the root locations may still have the wrong sign or end behavior.
Continue
Rational functions combine polynomial numerator and denominator factors. Their graphs need a domain analysis before the same factor-sign reasoning can be applied safely.
Apply · explain · revise
Practice and reasoning
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Enter the distinct real zeros of P(x)=−2(x+1)²(x−3)³.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
What is the degree of P(x)=5(x−2)⁴(x+3)²(x−1)?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Find P(0) for P(x)=−2(x+1)²(x−3)³.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For a degree-5 polynomial with negative leading coefficient, which end behavior is correct?
For P(x)=(x−1)²(x+2), give the sign on each interval separated by −2 and 1. Explain which root crosses the axis and why.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Factor P(x)=x³−2x²+x−2 over the reals and describe all complex roots with multiplicity. Explain why degree 3 does not imply three real graph crossings.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
