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AP Precalculus: Polynomial Functions

Master roots, theorems, end behavior, and graph analysis for polynomial functions

🔢 Roots & Zeros 📜 Key Theorems 📊 End Behavior 🎯 Graph Analysis

📚 Understanding Polynomial Functions

Polynomial functions are building blocks of calculus and appear throughout mathematics. This guide covers how to find and use roots, essential theorems for solving polynomials, end behavior patterns, and techniques for matching equations to graphs — all crucial skills for AP Precalculus success.

1 Roots / Zeros of Factored Polynomials

The roots (or zeros) of a polynomial are the values of \(x\) that make \(f(x) = 0\). When a polynomial is in factored form, these values are immediately visible.

Factored Form If \(f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n)\)
Then roots are: \(x = r_1, r_2, \ldots, r_n\)

Finding Roots from Factored Form

  • Set each factor equal to zero: \((x - r_k) = 0\)
  • Solve: \(x = r_k\)
  • The leading coefficient \(a\) affects shape but not the location of roots
  • Each root is an x-intercept on the graph
📌 Example

Find the roots of: \(f(x) = 2(x + 3)(x - 1)(x - 5)\)

Set each factor to zero:

\(x + 3 = 0 \Rightarrow x = -3\)

\(x - 1 = 0 \Rightarrow x = 1\)

\(x - 5 = 0 \Rightarrow x = 5\)

Roots: \(x = -3, 1, 5\)

2 Writing Polynomials from Roots

Given the roots of a polynomial, you can write its equation by creating factors from each root and multiplying them together.

General Form from Roots Given roots \(r_1, r_2, \ldots, r_n\):
\[f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n)\]

Steps to Write a Polynomial

  1. Write a factor \((x - r)\) for each root \(r\)
  2. If a root has multiplicity \(m\), write \((x - r)^m\)
  3. Include the leading coefficient \(a\) (use given point to find \(a\) if needed)
  4. Multiply out if standard form is required
📌 Example

Write a polynomial with roots \(x = -2, 0, 3\) and leading coefficient 4:

\(f(x) = 4(x - (-2))(x - 0)(x - 3)\)

\(f(x) = 4(x + 2)(x)(x - 3)\)

\(f(x) = 4x(x + 2)(x - 3)\)

Expanded: \(f(x) = 4x^3 - 4x^2 - 24x\)

💡 Finding the Leading Coefficient

If given a point \((x_0, y_0)\) on the graph, substitute it into the factored form and solve for \(a\).

3 Rational Root Theorem

The Rational Root Theorem provides a list of all possible rational roots of a polynomial with integer coefficients. Not all possibilities are actual roots — you must test them!

Possible Rational Roots \[\text{Possible roots} = \pm \frac{p}{q}\] where \(p\) = factors of constant term, \(q\) = factors of leading coefficient

Steps to Apply the Theorem

  1. List all factors of the constant term \(a_0\) (including \(\pm\))
  2. List all factors of the leading coefficient \(a_n\) (including \(\pm\))
  3. Form all possible fractions \(\pm \frac{p}{q}\)
  4. Test each candidate using synthetic division or direct substitution
  5. A remainder of 0 confirms it's a root
📌 Example

Find possible rational roots of: \(f(x) = 2x^3 + 5x^2 - 4x - 3\)

Constant term: \(-3\) → factors: \(\pm 1, \pm 3\)

Leading coefficient: \(2\) → factors: \(\pm 1, \pm 2\)

Possible rational roots: \(\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}\)

Test each to find actual roots.

⚠️ Important Limitations

The Rational Root Theorem only finds rational roots. Irrational roots (like \(\sqrt{2}\)) and complex roots won't appear in the list.

4 Complex Conjugate Theorem

The Complex Conjugate Theorem states that for polynomials with real coefficients, complex roots always come in conjugate pairs.

Conjugate Pair Rule If \(a + bi\) is a root (where \(b \neq 0\)),
then \(a - bi\) is also a root.

Key Implications

  • Complex roots come in pairs, so polynomials of odd degree always have at least one real root
  • If \(3 + 2i\) is a root, \(3 - 2i\) must also be a root
  • The product \((x - (a+bi))(x - (a-bi)) = x^2 - 2ax + (a^2 + b^2)\) gives a real quadratic factor
📌 Example

A polynomial has roots \(2, -1, 3+i\). What are all roots?

By the Complex Conjugate Theorem, if \(3 + i\) is a root, then \(3 - i\) is also a root.

All roots: \(2, -1, 3+i, 3-i\)

The polynomial has degree 4.

5 Irrational Conjugate Theorem

For polynomials with rational coefficients, irrational roots involving square roots also come in conjugate pairs.

Irrational Conjugate Rule If \(a + \sqrt{b}\) is a root (where \(\sqrt{b}\) is irrational),
then \(a - \sqrt{b}\) is also a root.
📌 Example

A polynomial with rational coefficients has root \(2 + \sqrt{5}\). What other root is guaranteed?

By the Irrational Conjugate Theorem: \(2 - \sqrt{5}\) is also a root.

💡 When Does This Apply?

This theorem requires rational coefficients. If a polynomial has irrational coefficients, irrational and complex roots don't necessarily come in pairs.

6 Descartes' Rule of Signs

Descartes' Rule of Signs uses sign changes in the polynomial's coefficients to predict the number of positive and negative real roots.

Positive Real Roots

Count sign changes in \(f(x)\).
Number of positive roots = sign changes OR less by a multiple of 2

Negative Real Roots

Count sign changes in \(f(-x)\).
Number of negative roots = sign changes OR less by a multiple of 2

📌 Example

Apply Descartes' Rule to: \(f(x) = x^4 - 3x^3 + 2x^2 + x - 5\)

\(f(x)\) signs: \(+, -, +, +, -\) → 3 sign changes

Positive roots: 3 or 1

\(f(-x) = x^4 + 3x^3 + 2x^2 - x - 5\)

\(f(-x)\) signs: \(+, +, +, -, -\) → 1 sign change

Negative roots: 1

7 Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra guarantees that every non-constant polynomial has at least one complex root, and a polynomial of degree \(n\) has exactly \(n\) roots when counting multiplicities.

The Fundamental Theorem A polynomial of degree \(n\) has exactly \(n\) complex roots
(counting multiplicities)

Key Implications

  • A degree 3 polynomial has exactly 3 roots (real or complex)
  • A degree 4 polynomial has exactly 4 roots (real or complex)
  • Complex roots come in conjugate pairs, so degree 3 polynomials always have at least 1 real root
  • Counting multiplicities means a double root counts as 2
📌 Example

How many roots does \(f(x) = x^5 - 2x^4 + x^3 - 1\) have?

Degree = 5, so exactly 5 roots (real or complex, counting multiplicities)

Since degree is odd and coefficients are real, there's at least 1 real root.

8 End Behavior

End behavior describes what happens to \(f(x)\) as \(x\) approaches positive and negative infinity. It's determined entirely by the leading term \(a_nx^n\).

End Behavior Patterns

Even Degree, \(a > 0\)
↗ ↗
Both ends go UP
\(x \to \pm\infty: f(x) \to +\infty\)
Even Degree, \(a < 0\)
↘ ↘
Both ends go DOWN
\(x \to \pm\infty: f(x) \to -\infty\)
Odd Degree, \(a > 0\)
↙ ↗
Left DOWN, Right UP
\(x \to -\infty: f \to -\infty\), \(x \to +\infty: f \to +\infty\)
Odd Degree, \(a < 0\)
↗ ↙
Left UP, Right DOWN
\(x \to -\infty: f \to +\infty\), \(x \to +\infty: f \to -\infty\)
📌 Example

Describe the end behavior of: \(f(x) = -3x^5 + 2x^3 - x + 7\)

Leading term: \(-3x^5\)

Degree: 5 (odd), Leading coefficient: \(-3\) (negative)

End behavior: Left end goes UP, Right end goes DOWN

As \(x \to -\infty\), \(f(x) \to +\infty\); As \(x \to +\infty\), \(f(x) \to -\infty\)

9 Matching Polynomials to Graphs (Zeros & Multiplicity)

The behavior of a graph at each x-intercept depends on the multiplicity of that root — how many times the factor appears.

Multiplicity and Graph Behavior

Odd Multiplicity
📈✖️📉
Graph CROSSES the x-axis
(multiplicity 1, 3, 5, ...)
Even Multiplicity
📈↩️📈
Graph TOUCHES and turns around
(multiplicity 2, 4, 6, ...)

Steps to Match Graph to Equation

  1. Check end behavior: Match degree (even/odd) and leading coefficient sign
  2. Count x-intercepts: Each is a root; note multiplicity
  3. Observe behavior at intercepts: Cross (odd) or touch (even)
  4. Check y-intercept: This is \(f(0)\), the constant term
📌 Example

What can we say about \(f(x) = (x+2)^2(x-1)(x-4)\)?

Roots: \(x = -2\) (mult. 2), \(x = 1\) (mult. 1), \(x = 4\) (mult. 1)

At \(x = -2\): Graph touches (even multiplicity)

At \(x = 1, 4\): Graph crosses (odd multiplicity)

Degree: \(2 + 1 + 1 = 4\) (even), Leading coefficient positive → Both ends UP

10 Domain & Range of Polynomials

Polynomial functions are defined for all real numbers. Their range, however, depends on the degree and leading coefficient.

Domain

Always all real numbers: \((-\infty, \infty)\)
No restrictions — polynomials have no denominators or radicals

Range

Odd degree: \((-\infty, \infty)\)
Even degree: Has a minimum or maximum value

📌 Examples

\(f(x) = x^3 - 2x\): Domain: \((-\infty, \infty)\), Range: \((-\infty, \infty)\)

\(f(x) = x^2 - 4\): Domain: \((-\infty, \infty)\), Range: \([-4, \infty)\) (minimum at vertex)

\(f(x) = -2x^4 + 5\): Domain: \((-\infty, \infty)\), Range: \((-\infty, 5]\) (maximum of 5)

11 Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry. This classification is determined by testing \(f(-x)\).

Even Function
\(f(-x) = f(x)\)
Symmetric about y-axis
Examples: \(x^2, x^4, \cos(x)\)
Odd Function
\(f(-x) = -f(x)\)
Symmetric about origin
Examples: \(x^3, x^5, \sin(x)\)

Testing Polynomials

  • A polynomial is even if it contains only even powers of \(x\) (including constants)
  • A polynomial is odd if it contains only odd powers of \(x\)
  • If it has both even and odd powers, it's neither
📌 Examples

\(f(x) = x^4 - 3x^2 + 5\): Only even powers → EVEN

\(f(x) = x^5 - 2x^3 + x\): Only odd powers → ODD

\(f(x) = x^3 + x^2 - 1\): Mixed powers → NEITHER

💡 Quick Test

Substitute \(-x\) for \(x\) and simplify. If you get the original, it's even. If you get the negative of the original, it's odd. Otherwise, it's neither.

📋 Quick Reference: Key Theorems

Rational Root Theorem

Possible roots: \(\pm \frac{\text{factors of } a_0}{\text{factors of } a_n}\)

Complex Conjugate Theorem

If \(a+bi\) is a root, so is \(a-bi\)

Fundamental Theorem

Degree \(n\) → exactly \(n\) roots (counting multiplicities)

Multiplicity Rule

Odd → crosses; Even → touches

Even Function

\(f(-x) = f(x)\) (y-axis symmetry)

Odd Function

\(f(-x) = -f(x)\) (origin symmetry)

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