Writing a Polynomial Function From Its Zeros

Every method in this chapter so far has started from a polynomial and worked toward its zeros. The reverse problem — construct a polynomial function with the stated properties — turns out to be easier, because the factor theorem does almost all of the work: each zero hands you a factor , and multiplying the factors together gives a polynomial with exactly those zeros. The only thing the zeros don’t decide is the constant in front, and that one gap is where every variation of the question lives.

From Zeros to Factors

The factor theorem says that is a zero of exactly when is a factor of . Read left to right it finds zeros; read right to left it builds polynomials. If the zeros are , then

has all of them as zeros, for any non-zero constant . Three things to be careful about when converting:

• The factor for a positive zero has a minus sign: zero gives .
• The factor for a negative zero has a plus sign: zero gives .
• A fractional zero like gives , which can be scaled to if you want integer coefficients — that scaling just moves a factor of into .

For zeros and :

With , . Check: and .

The Leading Coefficient Is Not Determined by the Zeros

This is the subtlety. , and all have zeros and , because multiplying by a constant scales the graph vertically without moving where it crosses the axis. So “the polynomial with zeros and ” is really a whole family, , and a question has to give you one extra fact to pick a member:

The “nothing extra” row is common in homework: form a polynomial whose zeros and degree are given. The expected answer is the version, but it does no harm to write and then set it to .

When the Degree Is Larger Than the Number of Zeros

If a question asks for a degree- polynomial but lists only three zeros, one of them has to be repeated. A zero of multiplicity contributes the factor , and the multiplicities must add up to the degree. The question will normally say which zero repeats (” is a zero of multiplicity ”); if it is a graph, the repeated zero is the one where the curve touches the axis and turns back instead of crossing — the zeros lesson covers how multiplicity shows in the shape.

If a question gives fewer constraints than the degree needs and does not say which zero repeats, there is no single answer — say so rather than guessing.

Complex Zeros Come in Pairs

A polynomial with real coefficients cannot have a lone complex zero. If is a zero, then its conjugate is a zero too. Multiply that pair of factors together first, because the cancels out and leaves a real quadratic:

That quadratic can then be multiplied into the remaining real factors with no complex arithmetic left. The shortcut for any conjugate pair :

The same trick works for irrational conjugates: a zero for a polynomial with rational coefficients drags in , and the pair multiplies to .

Worked Example: Form a Polynomial Whose Real Zeros and Degree Are Given

Form a polynomial of degree with zeros , and , and leading coefficient .

Step 1 — write one factor per zero, flipping each sign:

Step 2 — three zeros, degree : each is used once.

Step 3 — multiply two factors, then the third:

Step 4 — check one zero. . ✓

Worked Example: Fitting a Point to Find the Leading Coefficient

Find the polynomial of degree with zeros , and whose graph passes through .

Step 1 — set up the family:

Step 2 — substitute the point. At , must equal :

Step 3 — expand with . Note :

Check. . ✓

Worked Example: A Repeated Zero and a Complex Pair

Find a degree- polynomial with real coefficients that has zeros (multiplicity ) and .

Step 1 — list every zero with its multiplicity. forces its conjugate . So the zeros are — four zeros for degree . ✓

Step 2 — write the factors, pairing the conjugates:

Step 3 — multiply the conjugate pair first:

Step 4 — expand the rest:

With leading coefficient :

Every coefficient is real, as promised — that is what pairing the conjugates guarantees.

Write an Equation for the Polynomial Graphed Below

When the information comes as a graph rather than a list, read the zeros off the -intercepts and the multiplicities off the shape, then fit the -intercept for :

1. Zeros: each -intercept is a zero.
2. Multiplicity: a straight crossing is odd multiplicity (take ); a touch-and-turn is even (take ); a flattened S-shaped crossing is .
3. Degree: add the multiplicities. Check it against the end behavior — both tails the same way means even degree.
4. Leading coefficient: substitute the -intercept (or any other labelled point) and solve for . Its sign should match the end behavior.

For example, a graph that crosses at , touches at , and has -intercept gives , and means . The degree is , so the tails should go opposite ways — and with , down on the left and up on the right.

Common Mistakes to Avoid

• Keeping the zero’s sign in the factor. Zero gives , not . The factor is minus the zero, always.
• Forgetting the leading coefficient. is the full answer; is one member of it. If the question gives a point or a leading coefficient, you must use it.
• Using a complex zero without its conjugate. A single factor leaves in the coefficients. With real coefficients required, has to come along.
• Multiplying a conjugate pair term by term. Use or the difference-of-squares route; expanding all four products invites sign errors with .
• Miscounting the degree. Multiplicities add. Zeros make a quartic, not a cubic.
• Not checking. Substituting one zero back into the expanded polynomial takes ten seconds and catches almost every expansion slip.

Practice Problems

Work each one before opening the answer.

Problem 1. Write a polynomial with zeros and and leading coefficient .

Problem 2. Form a polynomial of degree whose zeros are , and .

Problem 3. Find the quadratic with zeros and that passes through .

Problem 4. Write a degree- polynomial with zeros (multiplicity ) and , leading coefficient .

Problem 5. Find a polynomial with real coefficients, leading coefficient , and zeros and .

Problem 6. A polynomial graph crosses the -axis at and , touches it at without crossing, and has both tails pointing down. Write an equation for it, given that it passes through .

Writing a polynomial from its zeros is the factor theorem used as a construction tool, plus one extra fact to fix the leading coefficient. Going the other way — taking a polynomial apart to find its zeros — is the finding polynomial zeros lesson, and reading zeros and multiplicities off a curve is practised further in graphing polynomials.