Polynomial Functions and | undefined Unit 1 Study Guide

The Core Idea: Polynomial Functions and Complex Zeros

This topic expands our understanding of polynomial functions by introducing the complete set of numbers required to account for all of their zeros: the complex number system. The central concept is encapsulated by the Fundamental Theorem of Algebra, which guarantees that a polynomial of degree $n$ has precisely $n$ zeros, provided we include complex numbers and count zeros with multiplicity.

Previously, we may have encountered polynomials that did not appear to have as many zeros as their degree suggested because they had few, if any, x-intercepts on a graph. This topic resolves that discrepancy by showing that the "missing" zeros are non-real complex numbers. We will learn that for polynomials with real coefficients, these complex zeros always appear in conjugate pairs ( $a + bi$ and $a - bi$ ). This section provides the analytical tools, such as the Rational Zero Theorem and the Factor Theorem, to systematically find all possible rational, irrational, and complex zeros of any polynomial function.

Key Formulas/Rules/Theorems

The Fundamental Theorem of Algebra

This theorem is a foundational guarantee in algebra. It states that any polynomial function of degree $n > 0$ has at least one zero in the complex number system. This ensures that our search for zeros will never be fruitless.

The $n$ Zeros Theorem

An extension of the Fundamental Theorem of Algebra, this states that a polynomial function of degree $n > 0$ has exactly $n$ complex zeros. This count includes any repeated zeros according to their multiplicity. For example, a polynomial of degree 5 will have exactly 5 complex zeros. Some may be real, some may be non-real complex, and some may be repeated.

The Complex Conjugate Theorem

This theorem applies specifically to polynomials with real coefficients. It states that if a complex number $a + bi$ (where $b \neq = 0$ ) is a zero of the polynomial, then its conjugate, $a - bi$ , must also be a zero. Non-real zeros always come in these pairs for polynomials with real coefficients.

The Rational Zero Theorem

This theorem provides a complete list of all possible rational zeros for a polynomial with integer coefficients. For a polynomial $P (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ , any rational zero must be of the form $\frac{p}{q}$ , where:

$p$ is an integer factor of the constant term, $a_{0}$ .
$q$ is an integer factor of the leading coefficient, $a_{n}$ .

The Remainder Theorem

This theorem creates a direct link between dividing a polynomial and evaluating it. It states that if a polynomial $p (x)$ is divided by the linear factor $(x - k)$ , the remainder of that division is equal to the value of the function at $k$ , or $p (k)$ .

$Remainder of \frac{p ( x )}{x - k} = p (k)$

The Factor Theorem

This theorem is a direct consequence of the Remainder Theorem and is crucial for factoring polynomials. It states that a number $k$ is a zero of a polynomial $p (x)$ if and only if $(x - k)$ is a factor of $p (x)$ . This works because if $(x - k)$ is a factor, the division has a remainder of 0. By the Remainder Theorem, this means $p (k) = 0$ , which is the definition of a zero.

Understanding the Relationship Between Zeros, Factors, and Coefficients

The key nuance of this topic is understanding the intricate web of connections between a polynomial's zeros, its factors, and its coefficients. The theorems provided are not isolated rules but a toolkit for moving between these different representations of the function.

The Factor Theorem is the bridge: knowing a zero $k$ immediately gives you a linear factor $(x - k)$ . The Complex Conjugate Theorem adds a condition: if your polynomial has real coefficients (as most do in this course) and you know $a + bi$ is a zero, you automatically know $a - bi$ is also a zero. This means you have found two zeros and thus two factors: $(x - (a + bi))$ and $(x - (a - bi))$ . When these two factors are multiplied together, the result is always a quadratic polynomial with real coefficients: $x^{2} - 2 a x + (a^{2} + b^{2})$ . This is why a polynomial can be built entirely from real coefficients yet still have non-real complex zeros.

The Rational Zero Theorem acts as a starting point for analysis. It doesn't find zeros for you, but it narrows down an infinite number of possibilities to a finite, testable list of potential rational zeros. You can then use the Remainder Theorem (often through synthetic division) to efficiently test these candidates. Once you find a zero $k$ , you use the Factor Theorem to divide the polynomial by $(x - k)$ , resulting in a simpler polynomial of a lower degree. This process is repeated until you are left with a quadratic factor, which can then be solved using the quadratic formula to find any remaining irrational or complex zeros.

Core Concepts & Rules

A polynomial function of degree $n$ will always have exactly $n$ zeros in the complex number system, when counting multiplicities.
For a polynomial with real coefficients, if $a + bi$ is a zero, then its conjugate $a - bi$ must also be a zero.
Non-real and irrational zeros of polynomials with real coefficients always appear in conjugate pairs.
The Rational Zero Theorem generates a list of all possible rational zeros by taking factors of the constant term and dividing them by factors of the leading coefficient.
The value of $p (k)$ is equal to the remainder when the polynomial $p (x)$ is divided by $(x - k)$ .
The number $k$ is a zero of the polynomial $p (x)$ if and only if $p (k) = 0$ .
The expression $(x - k)$ is a factor of the polynomial $p (x)$ if and only if $k$ is a zero.

Step-by-Step Example 1: Finding All Zeros of a Polynomial

Problem: Find all complex zeros of the polynomial function $P (x) = x^{4} - 3 x^{3} + 6 x^{2} - 12 x + 8$ .

Step 1: Use the Rational Zero Theorem to list possible rational zeros.

The constant term is $a_{0} = 8$ and the leading coefficient is $a_{n} = 1$ .

Factors of $p = 8$ : $\pm 1, \pm 2, \pm 4, \pm 8$
Factors of $q = 1$ : $\pm 1$
Possible rational zeros $\frac{p}{q}$ : $\pm 1, \pm 2, \pm 4, \pm 8$

Step 2: Test the possible zeros using the Remainder Theorem.

Let's test $x = 1$ :

$P (1) = (1)^{4} - 3 (1)^{3} + 6 (1)^{2} - 12 (1) + 8 = 1 - 3 + 6 - 12 + 8 = 0$

Since $P (1) = 0$ , $x = 1$ is a zero.

Step 3: Use the Factor Theorem and polynomial division.

Since $x = 1$ is a zero, $(x - 1)$ is a factor. We use synthetic division to divide $P (x)$ by $(x - 1)$ .

$1∣1 - 36 - 128∣1 - 24 - 8 - - - - - - - - - - - - - - - - - - - - 1 - 24 - 80$

The resulting quotient is $x^{3} - 2 x^{2} + 4 x - 8$ . So, $P (x) = (x - 1) (x^{3} - 2 x^{2} + 4 x - 8)$ .

Step 4: Repeat the process for the new, lower-degree polynomial.

Let $Q (x) = x^{3} - 2 x^{2} + 4 x - 8$ . We can test our list of possible rational zeros again. Let's test $x = 2$ :

$Q (2) = (2)^{3} - 2 (2)^{2} + 4 (2) - 8 = 8 - 8 + 8 - 8 = 0$

Since $Q (2) = 0$ , $x = 2$ is a zero. Therefore, $(x - 2)$ is a factor of $Q (x)$ .

Let's use synthetic division on $Q (x)$ :

$2∣1 - 24 - 8∣208 - - - - - - - - - - - - - - - - 1040$

The resulting quotient is $x^{2} + 0 x + 4 = x^{2} + 4$ .

Step 5: Solve the remaining quadratic factor.

We are left with the factor $x^{2} + 4$ . To find the zeros, we set it equal to zero:

$x^{2} + 4 = 0$

$x^{2} = - 4$

$x = \pm - 4$

$x = \pm 2 i$

So, the remaining zeros are $x = 2 i$ and $x = - 2 i$ .

Step 6: List all $n$ zeros.

The polynomial was degree 4, and we have found 4 zeros.

The zeros of $P (x)$ are: $1, 2, 2 i, - 2 i$ .

Step-by-Step Example 2: Constructing a Polynomial from its Zeros

Problem: Find a polynomial function $f (x)$ of the lowest possible degree with real coefficients, given that $x = 4$ and $x = 2 - 3 i$ are zeros, and $f (0) = 130$ .

Step 1: Identify all necessary zeros using the Complex Conjugate Theorem.

The problem states the polynomial has real coefficients. We are given a non-real complex zero, $2 - 3 i$ . Therefore, its conjugate, $2 + 3 i$ , must also be a zero.

The complete list of zeros for the lowest degree polynomial is: $4, 2 - 3 i, 2 + 3 i$ .

Step 2: Convert each zero into a linear factor.

Zero $k = 4$ corresponds to the factor $(x - 4)$ .
Zero $k = 2 - 3 i$ corresponds to the factor $(x - (2 - 3 i)) = (x - 2 + 3 i)$ .
Zero $k = 2 + 3 i$ corresponds to the factor $(x - (2 + 3 i)) = (x - 2 - 3 i)$ .

Step 3: Multiply the factors together, starting with the complex conjugate pair.

Multiplying the conjugate factors will result in a quadratic with real coefficients.

$(x - 2 + 3 i) (x - 2 - 3 i)$

Group the real and imaginary parts: $((x - 2) + 3 i) ((x - 2) - 3 i)$

This is in the form $(A + B) (A - B) = A^{2} - B^{2}$ , where $A = (x - 2)$ and $B = 3 i$ .

$(x - 2)^{2} - (3 i)^{2} = (x^{2} - 4 x + 4) - (9 i^{2}) = (x^{2} - 4 x + 4) - (9 (- 1)) = x^{2} - 4 x + 5$

Step 4: Multiply the result by the remaining factor.

Now, multiply this quadratic by the factor $(x - 4)$ .

$(x - 4) (x^{2} - 4 x + 5) = x (x^{2} - 4 x + 5) - 4 (x^{2} - 4 x + 5)$

$= (x^{3} - 4 x^{2} + 5 x) - (4 x^{2} - 16 x + 20)$

$= x^{3} - 4 x^{2} + 5 x - 4 x^{2} + 16 x - 20$

$= x^{3} - 8 x^{2} + 21 x - 20$

This gives us a family of polynomials $f (x) = a (x^{3} - 8 x^{2} + 21 x - 20)$ where $a$ is any non-zero real constant.

Step 5: Use the given point to find the specific value of $a$ .

We are given that $f (0) = 130$ .

$f (0) = a ((0)^{3} - 8 (0)^{2} + 21 (0) - 20) = a (- 20)$

Set this equal to 130:

$- 20 a = 130$

$a = \frac{130}{- 20} = - \frac{13}{2}$

Step 6: Write the final polynomial function.

Substitute $a = - \frac{13}{2}$ back into the function.

$f (x) = - \frac{13}{2} (x^{3} - 8 x^{2} + 21 x - 20)$

$f (x) = - \frac{13}{2} x^{3} + 52 x^{2} - \frac{273}{2} x + 130$

Using Your Calculator

The concepts in this topic are primarily analytical and are tested without a calculator. You must be able to apply the Rational Zero, Remainder, Factor, and Complex Conjugate theorems by hand. A calculator is best used as a tool to verify your work or to guide your initial guesses.

1. Verifying Real Zeros:

You can graph the polynomial function $y = P (x)$ to visually identify its x-intercepts. These are the real zeros. If the graph crosses the x-axis at $x = 2$ , this confirms that $2$ is a zero and $(x - 2)$ is a factor. This can help you choose which number from your Rational Zero Theorem list to test first.

Press [Y=] and enter the function.
Press [GRAPH]. - Use the $[CALC](2nd -> TRACE) menu and select2:zero` to find the precise values of the x-intercepts.

2. Verifying the Remainder Theorem:

The Remainder Theorem states that the remainder of $P (x) \div (x - k)$ is $P (k)$ . You can check this quickly on a calculator.

To find $P (3)$ for $P (x) = x^{3} - 5 x + 10$ :
On the home screen, type $3^{3} - 5 * 3 + 10$ and press [ENTER]. The result is the remainder.
Alternatively, after graphing the function, use the $[C A L C]$ menu and select 1:value. Enter `x=3$ to find the corresponding $y$ -value, which is $P (3)$ .

Important Note: A standard graphing calculator will not find non-real complex zeros for you. The process of finding complex zeros must be done analytically after factoring out all real zeros.

AP Exam Quick Hit

Common Question Types

Finding All Zeros: You are given a polynomial of degree 3 or 4, like $f (x) = x^{3} - 5 x^{2} + 17 x - 13$ , and asked to find all complex zeros. You would use the Rational Zero Theorem to find the real zero $x = 1$ , then use the quadratic formula on the remaining factor to find the complex zeros $2 \pm 3 i$ .
Constructing a Polynomial: You are told a polynomial with real coefficients has zeros $x = - 2$ and $x = 3 i$ . You must recognize that $x = - 3 i$ is also a zero, construct the factors $(x + 2)$ , $(x - 3 i)$ , and $(x + 3 i)$ , and multiply them to get $f (x) = x^{3} + 2 x^{2} + 9 x + 18$ .
Conceptual Theorem Application: A multiple-choice question might state: "The polynomial $p (x)$ is divided by $(x + 3)$ , and the remainder is 5. What is the value of $p (- 3)$ ?" Based on the Remainder Theorem, the answer is $p (- 3) = 5$ .

Common Mistakes

Forgetting the Conjugate Zero: The most common mistake. If a polynomial has real coefficients and $4 - i$ is a zero, students often forget that $4 + i$ must also be a zero.
Errors in the Rational Zero Theorem: Incorrectly forming the list of possible rational zeros. Remember, it is $\frac{factors of constant term}{factors of leading coefficient}$ . A simple mix-up here makes finding the first zero impossible.
Sign Errors with Factors: Confusing a zero $k$ with its factor $(x - k)$ . For example, if the zero is $x = - 5$ , the corresponding factor is $(x - (- 5)) = (x + 5)$ , not $(x - 5)$ .
Algebraic Errors with Complex Numbers: Making mistakes when multiplying factors involving complex numbers. A common error is calculating $(bi)^{2}$ as $b^{2}$ $ instead of $- b^{2}$ . For example, $(2 i)^{2} = 4 i^{2} = - 4$ .
Stopping After Finding Real Zeros: After finding one or two real zeros and reducing the polynomial to a quadratic, students sometimes stop without solving that final quadratic factor to find the remaining irrational or complex zeros. A degree $n$ polynomial must have $n$ zeros.

Polynomial Functions and Complex Zeros - AP PreCalculus Study Guide