Math Practice
Given the zeros, build the polynomial: each zero becomes a factor, the factors multiply, and a leading coefficient sits out front. These twelve problems run from two integer zeros up to quartics with a repeated zero, a complex pair, and a point to fit.
Write the polynomial with zeros
Factors
Write the quadratic with zeros
Both zeros are negative, so both factors carry a plus:
Write the polynomial with zeros
The zero
Form a polynomial of degree
The quadratic with zeros
Write the polynomial with zeros
Write a degree-
A polynomial with real coefficients has a zero at
Complex zeros of a real-coefficient polynomial come in conjugate pairs, so
Write the quadratic with real coefficients and leading coefficient
Use
Find the degree-
A cubic has zeros
A graph crosses the
A touch means an even multiplicity, so
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The writing a polynomial from its zeros lesson is one formula and one extra fact. These twelve problems apply it to integer zeros, repeated zeros, conjugate pairs, and graphs described in words.
One factor per zero, with a zero of multiplicity
Substitute the point’s
For
A crossing is a simple zero; a touch is multiplicity
Turn each zero r into the factor (x − r), multiply the factors, and multiply by a leading coefficient a. With no extra information, take a = 1.
Because scaling a polynomial by a constant does not move its zeros — x² − 2x − 15 and 4x² − 8x − 60 have the same zeros. A point on the graph, a stated leading coefficient, or the y-intercept is what pins down the specific polynomial.
Use one factor per zero, with a zero repeated as many times as its multiplicity so that the multiplicities add up to the given degree. If nothing else is stated, take the leading coefficient to be 1.
For a polynomial with real coefficients, yes: if 2 + i is a zero then 2 − i is too. Multiply the conjugate pair first using (x − p)² + q², which gives a real quadratic.
Each x-intercept is a zero. A crossing is a simple zero, a touch-and-turn is a zero of even multiplicity (usually 2), and a flattened S-crossing is multiplicity 3. Then use a labelled point, usually the y-intercept, to find a.
Substitute one of the zeros into the expanded polynomial — it should give 0. If a point was given, check that too.