A polynomial has a factor of (x − 3)² and a factor of (x + 1). Which statement correctly describes the graph's behavior at x = 3 and x = −1?
AThe graph crosses the x-axis at both x = 3 and x = −1, since both are zeros
BThe graph bounces off the x-axis at x = 3 and crosses at x = −1, because the multiplicity of 3 is even and the multiplicity of −1 is odd
CThe graph crosses the x-axis at x = 3 and bounces at x = −1, since (x − 3)² is a larger factor
DThe graph touches but does not cross at either zero, since both factors are squared in the expanded polynomial
Multiplicity determines the crossing behavior at each zero. The factor (x − 3)² gives x = 3 a multiplicity of 2 (even), so the graph touches the x-axis there and turns back — it bounces. The factor (x + 1) gives x = −1 a multiplicity of 1 (odd), so the graph crosses the x-axis there, changing sign. The rule is: even multiplicity → bounce (the factor squared is always non-negative, so the function doesn't change sign); odd multiplicity → cross (the factor changes sign as x passes through the zero). Treating all zeros the same way — always crossing — is the most common graphing error.
Question 2 Multiple Choice
A polynomial has a negative leading coefficient and an odd degree. Which end behavior is correct?
AThe graph rises on both the left and right sides
BThe graph falls on both the left and right sides
CThe graph rises on the left and falls on the right
DThe graph falls on the left and rises on the right
End behavior is determined entirely by the leading term. For odd-degree polynomials with a positive leading coefficient, the graph falls left and rises right (like y = x³). With a negative leading coefficient, the behavior flips: the graph rises left and falls right (like y = −x³). For large |x|, the leading term dominates all others — all the lower-degree terms become negligible. Sketching the end behavior first 'sets the frame' of the graph before filling in zeros, turning points, and the y-intercept.
Question 3 True / False
A degree-6 polynomial typically has exactly 5 turning points.
TTrue
FFalse
Answer: False
A degree-n polynomial has *at most* n − 1 turning points — this is a maximum, not a guarantee. A degree-6 polynomial can have 5 turning points, but it might have only 3, or even 1. The actual number of turning points depends on the specific polynomial and its zeros and their multiplicities. For example, a polynomial with a zero of multiplicity 6 at the origin (y = x⁶) has only one turning point (the vertex). Confusing 'at most n − 1' with 'exactly n − 1' is a common misconception.
Question 4 True / False
If a polynomial has a zero at x = 4 with even multiplicity, the graph touches the x-axis at x = 4 without crossing it.
TTrue
FFalse
Answer: True
When a zero has even multiplicity, the corresponding factor appears an even number of times — for example, (x − 4)². An even power is always non-negative regardless of whether x is slightly less than or slightly greater than 4. This means the polynomial does not change sign at x = 4: it approaches 0 from positive values (if the leading effect is positive), touches the axis, and returns to positive values. The graph 'bounces' like a parabola whose vertex sits on the x-axis. Odd multiplicity produces a sign change — crossing — because an odd power does change sign as x passes through the zero.
Question 5 Short Answer
Why can you sketch an accurate graph of a polynomial function from just three features — end behavior, zeros with multiplicities, and the y-intercept — without plotting many individual points?
Think about your answer, then reveal below.
Model answer: These three features fully constrain the shape of the graph between its anchor points. End behavior tells you where the graph heads as x goes to ±∞, establishing the tails. Zeros with multiplicities tell you exactly where the graph intersects or touches the x-axis and whether it crosses (odd multiplicity) or bounces (even multiplicity). Sign analysis between consecutive zeros tells you whether the curve is above or below the x-axis in each interval. The y-intercept provides one concrete point at x = 0. Because polynomial graphs are smooth and continuous — no sharp corners or breaks — connecting these anchor points while respecting end behavior and sign gives an accurate sketch. You are reading the polynomial's structure, not sampling it.
The deeper insight is that a polynomial's graph is entirely determined by its algebraic structure — degree, leading coefficient, and factored form — without any calculus. Each feature of the graph has a direct algebraic explanation, which is why reading the polynomial carefully is more powerful than plotting points.