The Student Success Commons at York County Community College: MAT 127: College Algebra

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Chapter 3.3

The zeros of a polynomial are the x-values that, when plugged in, make the polynomial equal to zero. Several theorems exist to aid in finding all the zeros of a polynomial:

1. Fundamental Theorem of Algebra:

• Statement: Every polynomial of degree n (where n > 0) has at least one complex zero.
• Implication: A polynomial of degree n has exactly n complex zeros, counting multiplicities (some zeros might be repeated; see definition of "Multiplicity" later in these notes). 

2. Factor Theorem:

• Statement:

  If k is a zero of a polynomial f(x), then (x - k) is a factor of f(x), and vice versa.

• Use:

  If you find a zero, you can use it to factor the polynomial and find other zeros. 

3. Rational Zero Theorem:

• Statement:

  If a polynomial f(x) with integer coefficients has a rational zero p/q (where p and q are integers with no common factors), then p must be a factor of the constant term of f(x), and q must be a factor of the leading coefficient.

• Use:

  Helps identify possible rational zeros to test using synthetic division or direct substitution. This is often a good place to start.

4. Remainder Theorem:

• Statement: If a polynomial f(x) is divided by (x - k), then the remainder is f(k). 
• Use: If f(k) = 0, then k is a zero of f(x), and (x - k) is a factor. 

5. Upper and Lower Bounds Theorem:

• Statement: If c > 0 and all the coefficients in the quotient and remainder are non-negative when dividing f(x) by (x - c), then c is an upper bound for the real zeros of f(x). If c < 0 and the coefficients in the quotient and remainder alternate signs, then c is a lower bound.
• Use: Helps narrow down the range of possible real zeros. 

6. Conjugate Zeros Theorem:

• Statement:

  If a polynomial with real coefficients has a complex zero a + bi, then its complex conjugate a - bi is also a zero.

• Use:

  Helps identify all zeros, including complex ones, if you know one complex zero. 

 

The multiplicity of a zero refers to the number of times a factor appears in the fully factored form of the polynomial. For example, if (x-a) appears twice in the factored form, then 'a' is a zero with a multiplicity of 2. 

 

Check out the following resources for more support:

• Intro to Zeros (video)
• Rational Zero Theorem Tested (video)
• Finding All the Zeros of a Polynomial (video)
• Finding Zeros of Polynomial Functions (article)
• Online Zeros Calculator (calculator)

Chapter 4.1

A one-to-one function is a function where each input maps to a unique output, meaning no two different inputs produce the same output. The horizontal line test is useful, visual way to determine if a function is or isn't one-to-one. Looking at the graph of f(x), if any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. (See "Horizontal Line Test and One-to-One Functions" video on YouTube.)

Inverse functions reverse the inputs and outputs of a function. That is, when you put an x value into the original function and get a y value, you can put that y value into the inverse function and get the original x value back. Note that not all functions have inverses; only functions that are one-to-one do, see below for more info. The notation for an inverse function "f^-1" (read as "f inverse"), where "f" represents the original function; meaning if you have a function named "f", its inverse would be written as f^-1. Keep in mind that the " -1 " does not represent an exponent, but rather signifies the inverse operation.

 

To find an inverse function:

1. Ensure the function is one-to-one
2. Replace f(x) with y so as to have an equation that begins "y = "
3. Swap x and y
4. Solve for y
5. Replace y with f^-1  

 

For example:

original equation:

f(x) = x + 5

replace f(x) with y:

y = x + 5

swap x and y:

x = y + 5

solve for y:

y = x - 5

replace y with f^-1(x): 

f^-1(x) = x - 5

You can verify one function is the inverse of another by using composition:

If f is one-to-one then g is the inverse of f if:

(f ○ g)(x) = x for every x in the domain of g

(g ○ f )(x) = x for every x in the domain of f

 

Check out the following resources for more support:

• Intro to Inverse Functions (article)
• Inverse Functions (article)
• Inverse Function--definition and examples (article)
• Finding Inverse Functions: linear (video example)