Student Success Commons: 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 4.2

An exponential function calculates the exponential growth or decay of data. The basic exponential function is f(x) = a^x, where x is the input variable as an exponent, and a is a positive number greater than 1.

Note: Exponential functions are useful for modeling real-world phenomena like populations, radioactive decay, interest rates, and the amount of medicine in the bloodstream!