Math · College algebra · Worked example
Sketch a polynomial from its zeros
Sketch f(x) = −(x + 2)(x − 1)²(x − 3): find its degree, end behavior, zeros with their multiplicities, and y-intercept.
Find the degree and leading coefficient
Multiplying the leading terms of the factors gives −x·x²·x = −x⁴: degree 4, leading coefficient −1. Expanding confirms it.
Read the end behavior
The degree is even and the leading coefficient negative, so both ends fall.
List the zeros
The zeros are −2 (multiplicity 1), 1 (multiplicity 2) and 3 (multiplicity 1). The graph crosses the axis at −2 and 3, and touches it at 1.
Find the y-intercept
Set x = 0.
Sketch
Rising from the lower left, the graph crosses at −2, passes through (0, 6), comes down to touch the axis at 1 and rises again, then turns, crosses at 3 and falls to the right. That makes three turning points, the most a degree-4 polynomial can have.
Result
Both ends fall; the graph crosses the x-axis at −2 and 3, touches it at 1, and meets the y-axis at 6.
Your turn
Describe the end behavior and the zeros of g(x) = x(x − 2)³.
Show the answer and explanation
Both ends rise; the graph crosses at 0, and crosses at 2 while flattening there.
The leading term is x·x³ = x⁴, degree 4 with a positive coefficient. The zero 0 has multiplicity 1 and the zero 2 has multiplicity 3; both are odd, so the graph crosses at each.
Keep exploring
Graph plots f with its zeros and y-intercept marked. Change the exponent on (x − 1) to 3 and watch the touch become a flattened crossing.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
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Graph the polynomial Check the expansion in Math Open worked example on a board Degree and leading term in Math ReferenceYour existing work stays on this device. Examples open as editable copies.