3.6 Zeros of Polynomial Functions

Polynomial Division and the Remainder Theorem

Polynomial division and the Remainder Theorem give you efficient ways to evaluate polynomials and find their zeros without doing tedious calculations every time. These techniques connect directly to factoring, graphing, and understanding the overall structure of polynomial functions.

Application of Remainder Theorem

The Remainder Theorem says that when you divide a polynomial $P(x)$ by a linear factor $(x - c)$, the remainder equals $P(c)$. Instead of performing full long division, you just plug $c$ into the polynomial.

• If $P(c) = 0$, the division is exact, meaning $(x - c)$ divides evenly into $P(x)$
• If $P(c) \neq 0$, that value is the remainder

Example: For $P(x) = x^3 - 4x^2 + 5x - 2$, find the remainder when dividing by $(x - 3)$:

$P(3) = 3^3 - 4(3)^2 + 5(3) - 2 = 27 - 36 + 15 - 2 = 4$

The remainder is 4, so $(x - 3)$ is not a factor of $P(x)$.

Polynomial long division or synthetic division is still useful when you need the full quotient, not just the remainder. Synthetic division is especially fast for dividing by linear factors.

Zeros of Polynomial Functions

Factor Theorem for Polynomial Zeros

The Factor Theorem is a direct consequence of the Remainder Theorem. It states:

• If $P(c) = 0$, then $c$ is a zero of $P(x)$ and $(x - c)$ is a factor
• Conversely, if $(x - c)$ is a factor of $P(x)$, then $P(c) = 0$

This gives you a two-way test: plug in a candidate value, and if the output is zero, you've found both a root and a factor.

Example: For $P(x) = x^3 - 3x^2 - 4x + 12$, test $c = 2$:

$P(2) = 2^3 - 3(2)^2 - 4(2) + 12 = 8 - 12 - 8 + 12 = 0$

Since $P(2) = 0$, we know 2 is a zero and $(x - 2)$ is a factor. You can then divide $P(x)$ by $(x - 2)$ to reduce the polynomial and find the remaining zeros.

Rational Zero Theorem Applications

The Rational Zero Theorem narrows down which rational numbers could be zeros of a polynomial with integer coefficients. Any rational zero must have the form:

$\pm \frac{p}{q}$

where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Steps to apply the Rational Zero Theorem:

1. List all factors of the constant term ($p$ values)
2. List all factors of the leading coefficient ($q$ values)
3. Form all possible fractions $\pm \frac{p}{q}$
4. Test each candidate using the Factor Theorem (plug it in or use synthetic division)
5. Once you find a zero, divide it out and repeat on the reduced polynomial

Example: For $P(x) = 3x^3 - 5x^2 - 11x + 15$:

• Factors of the constant term 15: $\pm 1, \pm 3, \pm 5, \pm 15$
• Factors of the leading coefficient 3: $\pm 1, \pm 3$
• Possible rational zeros: $\pm 1, \pm 3, \pm 5, \pm 15, \pm \frac{1}{3}, \pm \frac{5}{3}$

This list can look long, but it's finite. You then test these candidates systematically until you find actual zeros.

Classification of Polynomial Zeros

Zeros of a polynomial fall into distinct categories, and each type affects the graph differently.

By number type:

• Rational zeros can be found using the Rational Zero Theorem
• Irrational zeros often appear in conjugate pairs like $a + \sqrt{b}$ and $a - \sqrt{b}$
• Complex zeros always occur in conjugate pairs: if $a + bi$ is a zero, then $a - bi$ must also be a zero (for polynomials with real coefficients)

By multiplicity (how many times a zero is repeated as a factor):

• Multiplicity 1 (single zero): the graph crosses the x-axis at that point
• Multiplicity 2 (double zero): the graph touches the x-axis and turns around
• Odd multiplicity > 1: the graph crosses the x-axis but flattens out near the zero
• Even multiplicity > 1: the graph touches but does not cross

Example: $P(x) = (x - 2)(x - 3)^2(x - (1 + i))(x - (1 - i))$

• Single zero at $x = 2$ (graph crosses the x-axis)
• Double zero at $x = 3$ (graph touches and bounces off the x-axis)
• Complex zeros at $1 + i$ and $1 - i$ (no x-intercepts for these; they don't appear on the real graph)

When you've factored out all the real linear factors and are left with a quadratic, use the quadratic formula to find any remaining zeros, including complex ones.

Linear Factorization Theorem Usage

The Linear Factorization Theorem guarantees that every polynomial of degree $n$ can be written as a product of exactly $n$ linear factors (over the complex numbers), plus a leading coefficient:

$P(x) = a_n(x - c_1)(x - c_2) \cdots (x - c_n)$

In practice, when working over the reals, complex conjugate pairs combine into irreducible quadratic factors. For instance, the factors $(x - (a + bi))(x - (a - bi))$ multiply out to a quadratic with real coefficients.

Building a polynomial from given zeros:

1. Write a linear factor for each real zero
2. For zeros with multiplicity greater than 1, raise that factor to the appropriate power
3. For each complex zero, include its conjugate as well
4. Multiply all factors together

Example: Construct a polynomial with zeros $-1$, $2$ (multiplicity 3), and $3 - 2i$:

Since complex zeros come in conjugate pairs, $3 + 2i$ is also a zero.

$P(x) = (x + 1)(x - 2)^3(x - (3 - 2i))(x - (3 + 2i))$

The complex pair multiplies out to $(x^2 - 6x + 13)$, giving you a polynomial with all real coefficients. This polynomial has degree $1 + 3 + 2 = 6$.

Analyzing Polynomial Functions

Descartes' Rule of Signs

Descartes' Rule of Signs gives you an upper bound on the number of positive and negative real zeros before you start testing candidates.

For positive real zeros:

1. Write $P(x)$ in standard form (descending powers)
2. Count the number of sign changes between consecutive nonzero terms
3. The number of positive real zeros is that count, or less by an even number

For negative real zeros:

1. Compute $P(-x)$ (replace every $x$ with $-x$ and simplify)
2. Count sign changes in $P(-x)$
3. The number of negative real zeros is that count, or less by an even number

Example: $P(x) = 2x^4 - 3x^3 - 12x^2 + 5x + 8$

Signs of coefficients: $+, -, -, +, +$

Sign changes: $(+ \to -)$, $(- \to +)$ → 2 sign changes, so there are 2 or 0 positive real zeros.

Now compute $P(-x) = 2x^4 + 3x^3 - 12x^2 - 5x + 8$

Signs: $+, +, -, -, +$

Sign changes: $(+ \to -)$, $(- \to +)$ → wait, let's recount carefully: $(+, +)$ no change, $(+, -)$ change, $(-, -)$ no change, $(-, +)$ change → 2 sign changes, so there are 2 or 0 negative real zeros.

(Note: The original count of 3 sign changes for $P(-x)$ was incorrect. Substituting $-x$ gives $2x^4 + 3x^3 - 12x^2 - 5x + 8$, which has 2 sign changes, not 3.)

Polynomial Modeling in Real-World Problems

Polynomials can model physical situations where quantities depend on one variable raised to different powers.

General approach:

1. Identify the key variable and express other quantities in terms of it
2. Set up a polynomial equation from the given relationships
3. Solve using factoring, the Rational Zero Theorem, or synthetic division
4. Check that your solutions make sense in context (reject negative lengths, for instance)

Example: A rectangular prism has a square base and a volume of 250 cubic centimeters. The height is 3 cm more than the side length of the base. Find the dimensions.

• Let $x$ = side length of the base, so the height = $x + 3$
• Volume: $x^2(x + 3) = 250$
• Expanding: $x^3 + 3x^2 - 250 = 0$
• Use the Rational Zero Theorem to test candidates. Since $x$ represents a length, only positive real solutions are valid.

Graphing and Analyzing Polynomial Functions

The graph of a polynomial ties together everything from this section. Here's what to look for:

• x-intercepts correspond to real zeros. The behavior at each intercept depends on the zero's multiplicity (crosses vs. touches).
• y-intercept is the constant term, found by evaluating $P(0)$.
• End behavior depends on the degree and leading coefficient:
  • Even degree with positive leading coefficient: both ends go up
  • Even degree with negative leading coefficient: both ends go down
  • Odd degree with positive leading coefficient: falls left, rises right
  • Odd degree with negative leading coefficient: rises left, falls right

Combining end behavior, intercepts, and multiplicity information lets you sketch a reasonable graph without plotting dozens of points. Complex zeros don't produce x-intercepts, but they still count toward the total degree of the polynomial.