MHF4U Grade 12 Advanced Functions – Polynomial Equations and Inequalities Test

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
    • Remainder 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 a1,a2,a3,…an can be represented by an equation of the form:

y = k(x – a1)(x – a2)(x – a3) . . . (x – an), 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.