Section 4.3 - Zeros of Polynomials

Section Objectives

1. Apply the Fundamental Theorem of Algebra.
2. Use common tests to determine the nature of a polynomial's zeros.
3. Factor a polynomial with real coefficients into a product of linear factors and irreducible quadratic factors.

Rational Zeros Theorem

If $f(x)= a_n x^n+a_{n-1} x^{n-1}+⋯+a_1 x+a_0$ is a polynomial with integer coefficients, then any rational zero of $f$ must have the form $p/q$ where $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$.

Example

• List all possible rational zeros of $f(x)=2x^3+5x^2-4x-3$. (Solution)

To find the rational zeros of a polynomial with integer coefficients:

1. List all rational zeros.
2. If appropriate, use a graphing calculator to narrow done the list.
3. When very simple to do so, check a possible zero by directly evaluating the polynomial.
4. When synthetic division is more appropriate, check a possible zero by using synthetic division to evaluate the polynomial.

Example

• Find all the zeros of $f(x)=2x^3+5x^2-4x-3$. Then give a complete factorization of $f$. (Solution)

Some Other Useful Tests

Descartes' Rule of Signs: When a polynomial with real coefficients is written is order of decreasing exponents,

1. count the sign changes between nonzero coefficients. The number of positive, real zeros is that number or fewer by an even number.
2. change the signs of the odd-degree terms and then count the sign changes between nonzero coefficients. The number of negative, real zeros is that number or fewer by an even number.

Examples

• Apply Descartes' rules of signs: $\quad g(x)=x^3+2x^2-5x-6$ (Solution)

• Apply Descartes' rules of signs: $\quad h(x)=2x^4+7x^3+28x^2+112x-64$ (Solution)

Intermediate Value Theorem: If $f$ is a polynomial and $a$ and $b$ are two real numbers for which $f(a)$ and $f(b)$ have opposite signs, then $f$ has at least one real zero between $a$ and $b$.

Example

• Show that $f(x)=x^3+3x-7$ has at least one real zero between $1$ and $2$. (Solution)

Fundamental Theorem of Algebra

If $p$ is a polynomial of degree $n$, $n≥1$, then $p$ has at least one zero.

Some important results follow from the fundamental theorem:

• A polynomial of degree $n$ has exactly $n$ zeros, counting multiplicity.
• A polynomial of degree $n$ can be written in factored form as a product of $n$ linear factors.
• The graph of a polynomial of degree $n$ has at most $n$ $x$-intercepts and $n-1$ turning points.

Conjugate Roots Theorem: Suppose $p$ is a polynomial with real coefficients. If the complex number $a+bi$ is a zero of $p$, then the complex number $a-bi$ is also a zero of $p$. Thus the complex zeros of a polynomial with real coefficients always occur in conjugate pairs.

A quadratic factor of a polynomial with real coefficients is said to be irreducible if its zeros are imaginary complex conjugates.

Examples

• Use the quadratic formula to find the zeros of $g(x)=x^2-6x+13$. (Solution)

• Construct a 4th-degree polynomial with real coefficients with zeros 2, -5, and $1+i$ and whose graph passes through $(1,12)$. (Solution)

Factoring Polynomials

Finding all real and complex zeros of a polynomial is typically a difficult task. Here are some guidelines:

1. Start by factoring as much as is easily possible.
2. Use the quadratic formula if applicable.
3. Apply the Rational Zeros Theorem (or other tests).
4. Test the possible rational zeros and, each time you find one, use synthetic division to deflate the polynomial (divide out the corresponding linear factor).
5. Repeat steps 1-4, as necessary, on the deflated polynomial.
6. Use a graphing calculator to help locate the real zeros.
7. If necessary, apply more advanced zero-finding techniques.

Examples

• Completely factor the polynomial $f(x)=x^4-8x^3+200x-625$. (Solution)