**Key Terms**

- Factor Theorem – A number a is a root of a polynomial function if and only if, when dividing the polynomial by (x – a), the remainder equals zero.
- If you get a remainder when you divide polynomials, you do NOT have a root.

- Rational Root Theorem – The rational roots of a polynomial function F(x) can be written in the form , where p is a factor of the constant term of the polynomial, and q is a factor of the leading coefficient.
- Remainder Theorem – A theorem stating that when the polynomial in F(x) is divided by x – a, the remainder is F(a).
- Also, when F(x) is divided by x + a, the remainder is F(-a).

**Review**

- Divisibility Rules

- Roots
- Roots are the “zeros” of factors.
- Roots are the x-intercepts of the graph.
- To find the root of a factor, set the factor equal to zero.
- Ex. has factors of and
- Set the factors to equal zero and solve for x: and
- For the first factor: (x + 1), subtract 1 on both sides: x = -1
- For the second factor (x + 2), subtract 2 on both sides: x = -2
- Lastly, test the roots using substitution.
- For F(-1)
- So, -1 IS a root of the polynomial.

- For F(-2)
- So, -2 IS a root of the polynomial.

- Roots and Factors
- Roots have the opposite signs from their factors
- If (x + 4) is a factor, then x = -4 is a root.
- If (x – 6) is a factor, then x = 6 is a root.

- Roots have the opposite signs from their factors

- Factoring
- The constant term of a polynomial may have positive or negative factors
- The x-term of a polynomial is the sum of the factors of the constant term (which may be positive or negative).

**Notes**

- Factors, Roots, and Remainders
- A number a is a root of P(x) if and only if the remainder, when dividing the polynomial by x – a, equals zero.
- If a polynomial is divided by (x – a) and the remainder equals zero, then (x – a) is a factor of the polynomial.

- Roots
- If the leading coefficient is 1 (problem 2), just find all the factors of the constant term.
- If the leading coefficient is not 1 (problem 1), use the rational root theorem.
- The rational roots can be written as
- p is a factor of the constant term
- q is a factor of the leading coefficient

- Coefficient of 1
- Example:
- Possible roots are:

- Coefficient of a Number Other than 1
- Example:

- Conjugates
- How do the three rules below help you check your answers to polynomial equations?
- You know the maximum number of answers you should have.
- You know that for every radical and complex answer, you should also have its conjugate as another answer.

- How do the three rules below help you check your answers to polynomial equations?

- Complex Roots
- For graphs, there are NO x-intercepts for complex roots.

- Factoring Polynomials
- You only need to find one root!
- You don’t have to test
*all*the possible roots with synthetic division. - When you find one that works (in other words, when the remainder is zero), move on to step 3
- To begin factoring, start with the roots 1, then try -1

**Examples**

- Example 1
- What is a root (or zero) of a polynomial function?
- Answer: A value of the variable that makes the polynomial equal to zero

- Example 2
- -6 is NOT a root because there is a remainder of 11
- (x + 6) is NOT a factor because there is a remainder of 11
- When x = -6,
- When is divided by , the remainder is 11

- Example 3
- 4 is a root of
- is a factor of
- ÷ =
- ÷ =
- The remainder is zero