Find a polynomial P(x) of lowest degree, with leading...

Find a polynomial $P(x)$ of lowest degree, with leading coefficient $1,$ that has the indicated set of zeros. Leave the answer in a factored form. Indicate the degree of the polynomial. $i \sqrt{3} \text { (multiplicity } 2),-i \sqrt{3} \text { (multiplicity } 2),$ and 4 (multiplicity $3)$

Find a polynomial $P(x)$ of lowest degree, with leading coefficient $1,$ that has the indicated set of zeros. Leave the answer in a factored form. Indicate the degree of the polynomial.
$i \sqrt{3} \text { (multiplicity } 2),-i \sqrt{3} \text { (multiplicity } 2),$ and 4 (multiplicity $3)$

Key Concepts

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of the variable that appears with a nonzero coefficient. When a polynomial is expressed in its factored form, the degree is the sum of the exponents (multiplicities) of each factor. This concept is crucial for understanding the overall behavior and potential number of zeros of the polynomial.

Factored Form

Expressing a polynomial in factored form involves writing it as a product of its linear (or irreducible quadratic) factors, each raised to a power corresponding to the multiplicity of the zero it represents. This representation provides clear insight into the roots of the polynomial and their multiplicities.

Monic Polynomial

A monic polynomial is one in which the leading coefficient, the coefficient of the highest degree term, is equal to one. This normalization simplifies the form of the polynomial and is often a required condition when specifying a unique polynomial that meets given criteria.

Complex Conjugate Pairs

When a polynomial has real coefficients, any non-real complex zeros must occur in conjugate pairs. This means that if a + bi is a zero, then its complex conjugate a - bi is also a zero. This property ensures that the polynomial will have real coefficients when multiplied out, preserving the structure of the polynomial in the real number system.

Multiplicity

Multiplicity refers to the number of times a specific zero is repeated as a root of the polynomial. If a zero has multiplicity greater than one, the corresponding factor appears raised to that power. The multiplicity influences the behavior of the graph at the zero, such as whether the graph crosses or touches the x-axis.

Zeros of a Polynomial

The zeros of a polynomial are the values for which the polynomial equals zero. They correspond directly to the factors of the polynomial in its factored form. Knowing the zeros of a polynomial allows you to reconstruct the polynomial by writing a product of linear factors (or irreducible factors in the case of complex zeros) corresponding to each zero.

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