**Grade 12 – Advanced Functions – Polynomial Equations and Inequalities**

** **

**Remainder Theorem**

**Long Division**can be used to divide a polynomial by a binomial.- The result of the division of a polynomial function P(x) by a binomial of the form x – b can be written as P(x) = (x-b)Q(x) + R where Q(x) is the quotient and R is the remainder.
**Division Statement:**divisor x quotient + remainder = dividend- can be used to check the result of a division
- R
**emainder theorem**states that when a polynomial function P(x) is divided by x – b, the remainder is P(b), and when it is divided by ax-b, the remainder is P(b/a), where a and b are integers and a not = 0.

**Factor Theorem**

- For integer values of a and b, with a not equal 0,
**Factor Theorem**states that x – b is a factor of a polynomial P(x) if and only if P(b) = 0.- Similarly, if ax – b is a factor of P(x) if and only if P(b/a) = 0

**Integral Zero Theorem**states that if x – b is a factor of a polynomial function P(x) with leading coefficient 1 and remaining coefficients that are integers, then b is a factor of the constant term P(x).**Rational Zero Theorem**states that if P(x) is a polynomial function with integer coefficients and x = b/a is a rational zero of P(x), then- b is a factor of the constant term of P(x)
- a is a factor of the leading coefficient of P(x)
- ax – b is a factor of P(x)

** **

**Polynomial Equations**

**Real roots**of a polynomial equation P(x) = 0 correspond to the x-intercepts of the graph of the polynomial function P(x).**X-intercepts**of the graph of a polynomial function correspond to the real roots of the related polynomial equation.- If a polynomial equation is factorable, factoring the polynomial, setting its factors equal to zero, and solving each factor will determine the roots.

**Families of a Polynomial Function**

- A
**family of functions**is a set of functions with the same characteristics. - Polynomial functions with graphs that have the same x-intercepts belong to the same family.
- A family of polynomial functions with zeros a
_{1},a_{2},a_{3},…a_{n}can be represented by an equation of the form:

y = k(x – a_{1})(x – a_{2})(x – a_{3}) . . . (x – a_{n}), where k is a real number not equal to zero

- An equation for a particular member of a family of polynomial functions can be determined if a point on the graph is known.

** **

**Solving Inequalities or Inequations**

- A
**polynomial inequality**results when the equal sign in a polynomial equation is replaced with an inequality symbol. - The real zeros of a polynomial function, or x-intercepts of the corresponding graph, divide the x-axis into intervals that can be used to solve a polynomial inequality.
- Polynomial inequalities may be solved graphically by determining the x-intercepts and then using the graph to determine the intervals that satisfy the inequality.
- Factoring inequalities can be solved algebraically by:
- Considering all cases
- Using intervals and testing values in each interval
- Table and number lines can help organize intervals and to provide a visual clue to solutions.