Constructing Polynomials from Given Zeros

From Zeros to Factors

A polynomial can be described by its coefficients in standard form, but that form does not always make its zeros easy to see. For example, the expression \( (x-2)(x+1) \) immediately identifies the zeros \(2\) and \(-1\). This tutorial uses that connection in reverse: starting with real zeros and their multiplicities, we will construct a polynomial that has those features.

The key translation is simple but important. A zero \(r\) corresponds to the factor \(x-r\), not \(x+r\) unless \(r\) is negative. If the zero has multiplicity greater than one, its factor is repeated by raising it to that multiplicity. A nonzero constant in front then sets the polynomial's leading coefficient.

The nonzero condition matters: every power of \(x-r\) divides the zero polynomial, so it does not have a finite multiplicity in this sense. In what follows, each constructed polynomial has a nonzero leading coefficient and is therefore nonzero.

The Construction Rule

Suppose distinct real numbers \(r_1,\ldots,r_k\) are specified as zeros, with respective positive integer multiplicities \(m_1,\ldots,m_k\). The basic polynomial built from exactly these linear factors is

Here \(a\) is the leading coefficient. Each factor \(x-r_i\) is monic, meaning its leading coefficient is \(1\). Consequently, their product is also monic, and multiplying by \(a\) gives leading coefficient \(a\). The degree is the sum of the multiplicities:

This construction gives a polynomial with the specified zeros and multiplicities. It also uses no extra factors, so its real zeros are precisely the listed numbers. If a problem asks only for a polynomial that includes certain zeros, other choices may be possible: an additional polynomial factor can introduce more zeros or increase the degree. The product above is the direct construction when only the stated real zeros are wanted.

Why the Product Has the Required Properties

The construction works because a product is zero whenever one of its factors is zero. At \(x=r_i\), the factor \(x-r_i\) equals zero, so \(P(r_i)=0\). To check the multiplicity, separate that factor from all the others:

All the listed roots are distinct. Thus, when \(x=r_i\), every factor in \(Q_i(r_i)\) is nonzero: \(a\ne0\), and \(r_i-r_j\ne0\) whenever \(j\ne i\). Therefore \(Q_i(r_i)\ne0\). The factor \((x-r_i)^{m_i}\) divides \(P\), while one more factor \(x-r_i\) cannot divide it, because the remaining factor \(Q_i\) does not vanish at \(r_i\). This proves that the multiplicity is exactly \(m_i\).

The leading coefficient and degree follow directly from multiplying the leading terms. Each \((x-r_i)^{m_i}\) has leading term \(x^{m_i}\), so the product of those terms is \(x^{m_1+\cdots+m_k}\). Multiplication by \(a\) makes the leading term \(a x^{m_1+\cdots+m_k}\).

Worked Examples

What the Data Determine

Specified zeros and multiplicities determine the factors, but not by themselves the leading coefficient. For instance, multiplying the same product by \(2\) or by \(-5\) preserves its zeros and their multiplicities while changing its leading coefficient. The zero polynomial is not an option when exact multiplicities are specified, because multiplicity here is defined for nonzero polynomials.

When the zeros are stated to be all the zeros and the degree equals the sum of their multiplicities, the leading coefficient determines the polynomial uniquely. To see why, repeatedly use the Factor Theorem: if \(r\) is a zero of a polynomial, then \(x-r\) is a factor. Applying this for the specified multiplicities shows that the product of all the stated factors divides the polynomial. If the polynomial has degree equal to the degree of that product, the quotient must have degree zero, so it is a constant. The leading coefficient fixes that constant.

If only some zeros are specified and the degree is not fixed, there is no such uniqueness: a polynomial can contain those factors and also have additional factors. The direct construction is still valid, but it is one particular choice.

A Common Sign Error

A frequent mistake is to use \(x+r\) for a zero \(r\). If the desired zero is \(-2\), the incorrect factor \(x-2\) does not vanish there:

The correct factor is \(x-r\). With \(r=-2\), this becomes

A reliable check is to substitute the proposed zero into its factor. The factor must equal zero. After building the product, also check the exponents, the sum of multiplicities for the degree, and the constant multiplier for the leading coefficient.

Why This Representation Is Useful

The factored representation stores several kinds of information at once. It makes the specified zeros visible, records how many times each corresponding factor occurs, and gives the degree by adding the exponents. It also allows the leading coefficient to be read without expanding. Expansion can be useful when standard form is requested, but it is not necessary to establish these features.

When constructing a polynomial, begin with its required features rather than expanding prematurely. Build the factors first, attach the scale last, and verify each condition in the representation where it is easiest to see. This approach reduces sign errors and keeps the relationship between zeros, multiplicities, degree, and leading coefficient explicit.