3.6 The Real Zeros of a Polynomial Function: Remainder and Factor Theorems, Rational Zeros

The Real Zeros of a Polynomial Function

Graph of a Polynomial Function

A polynomial function is an expression of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. The graph of a polynomial function is continuous and smooth, with no breaks or sharp corners.

• Degree of the polynomial: The highest power of $x$ in the polynomial.

• Maximum number of real zeros: Equal to the degree $n$.

• Intercepts: Real zeros correspond to $x$-intercepts of the graph.

• Factoring: If $r$ is a real zero, then $(x - r)$ is a factor of $f(x)$.

Example: If $f(r) = 0$, then $r$ is an $x$-intercept and a solution to $f(x) = 0$.

Remainder Theorem

The Remainder Theorem states that for a polynomial $f(x)$, if $f(x)$ is divided by $(x - c)$, then the remainder is $f(c)$.

• Application: Use synthetic or long division to divide $f(x)$ by $(x - c)$ and evaluate the remainder.

Example: Find the remainder of $f(x) = x^3 - x^2 - 4x^2 + 5$ when divided by $x - 3$.

Factor Theorem

The Factor Theorem is a direct consequence of the Remainder Theorem:

• If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$.

• If $(x - c)$ is a factor of $f(x)$, then $f(c) = 0$.

Example: Use the Factor Theorem to determine whether the function $f(x) = x^3 + x^2 - 2x - 2$ has $x - 1$ or $x + 2$ as a factor.

Number of Real Zeros

A polynomial function of degree $n$ has at most $n$ real zeros. Each real zero corresponds to an $x$-intercept of the graph.

Rational Zeros Theorem

The Rational Zeros Theorem provides a method to list all possible rational zeros of a polynomial function with integer coefficients.

• Let $f(x) = a_nx^n + \ldots + a_0$.

• Any rational zero is of the form $\frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$.

Step	Description
1	List all factors of the constant term ($a_0$).
2	List all factors of the leading coefficient ($a_n$).
3	Form all possible fractions $\frac{p}{q}$ (positive and negative).

Example: List the potential rational zeros of $f(x) = 2x^3 + 3x^2 - 7x - 6$.

• Factors of $-6$ (constant term): $\pm1, \pm2, \pm3, \pm6$

• Factors of $2$(leading coefficient):$\pm1, \pm2$

• Possible rational zeros: $\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}$

Finding Real Zeros and Factoring

To find the real zeros of a polynomial:

1. List all possible rational zeros using the Rational Zeros Theorem.

2. Test each candidate using synthetic or long division, or by direct substitution.

3. Once a zero is found, factor out $(x - r)$ and repeat the process for the reduced polynomial.

Example: Find the real zeros of $f(x) = x^5 - x^4 - 4x^3 + 8x^2 - 32x + 48$ and write the function in factored form.

Summary Table: Theorems for Polynomial Zeros

Theorem	Statement	Application
Remainder Theorem	$f(x)$ divided by $(x - c)$ leaves remainder $f(c)$	Evaluate $f(c)$ to find remainder
Factor Theorem	If $f(c) = 0$, then $(x - c)$ is a factor	Test zeros and factor polynomials
Rational Zeros Theorem	Possible rational zeros are $\frac{p}{q}$	List and test candidates for zeros

Additional info:

• Long division and synthetic division are standard methods for dividing polynomials and testing zeros.

• Once all real zeros are found, the polynomial can be written as a product of linear and/or irreducible quadratic factors.