MHF4U Unit Outline
Chapter 2: Polynomial Equations and Inequalities
Throughout this chapter, you will build upon what you have learned about polynomial functions and go into more depth
about the algebraic models and manipulations behind their key features.
By the end of this chapter, you will be assessed on your ability to:
C1.0
C3.0
C4.0
identify and describe some key features of polynomial functions, and make connections between the numeric,
graphical, and algebraic representations of polynomial functions.
solve problems involving polynomial equations graphically and algebraically
demonstrate an understanding of solving polynomial inequalities
Section
2.1 The
Remainder
Theorem
Learning goals
How do you divide polynomials using long
division?
How can you determine a remainder without
dividing?
Vocab: remainder theorem, monomial,
binomial, trinomial, polynomial, remainder,
quotient, divisor, dividend, quotient form,
restrictions, placeholder
Key equations: P( x)  Q( x)  R where P(x)
ax  b
ax  b
is the polynomial being divided by ax – b and
where Q(x) is the quotient and R is the
remainder.
P( x)  (ax  b)Q( x)  R is the corresponding
statement that can be used to check the
division of P(x) by (ax-b).
b
when P(x) is divided by ax – b
P   R
Homework
Pg 91 #1 – 9, 20 optional
Pg 92 # 11, 13, 14, 16, 17 mandatory
Skills: perform long division of a polynomial by a binomial,
identify restrictions on the variable, write two equivalent
equations using the dividend, divisor, quotient and remainder
(the division statement and quotient form and the
corresponding statement that isolates the dividend), use the
remainder theorem to determine the remainder and use the
remainder theorem to solve for an unknown coefficient,
explain the use of a placeholder
a
2.2 The Factor
Theorem
How can you determine a factor of a
polynomial?
Vocab: factor, factor theorem, integral zero
theorem, rational zero theorem, dimensions,
non-factorable
Key equations: P( x)  Q( x) when ax – b is a
ax  b
factor of P(x)
P( x)  (ax  b)Q( x)
is the corresponding
statement that can be used to check the
division.
b
when P(x) is divided by ax – b
P   0
Pg 102 #1 – 5, 6def, 14 optional
Pg 102 #8, 11, 12, 13, 15 mandatory
Skills: use the factor theorem (special case of the remainder
theorem) to determine factors of a polynomial, use the
rational zero theorem or integral zero theorem (special case
of the rational zero theorem) to identify possible factors to
test with the remainder (factor) theorem, solve problems
using a combination of the rational zero theorem and the
remainder (factor) theorem, factor quadrilaterals by
inspection, decomposition or the quadratic formula, rewrite
polynomial equations in factored form
a
2.3 Polynomial
Equations
Factorable polynomial functions can be
rewritten in the form
f(x) = k (x – a1) (x – a2) (x – a3)… where the
zeros are a1, a2, a3 … and k is a real number
not equal to 0
How are roots, x-intercepts, and zeros
related?
Vocab: real and non-real roots, rational and
irrational root, x-intercepts, zeros, nonfactorable
Pg 110 #1 – 4, 9 optional
Pg 111 # 5, 8, 11, 12, 14, 16 mandatory
Skills: solve polynomial equations by factoring using
common factoring, factoring a quadratic, factor by grouping,
using the rational zero theorem and division, identifying real
and non-real roots, explaining the connection between roots
of an equation, x-intercepts of a graph and zeros of a
function, solve problems by determining zeros of a
polynomial function including those that are not factorable
(Continued next page )
2.4 Families of
Polynomial
Functions
How are polynomial functions with the same
zeros related?
Vocab: families of polynomial functions
Key equation: Families of polynomial
functions can be rewritten in the form
f(x) = k (x – a1) (x – a2) (x – a3)… where the
zeros are a1, a2, a3 … and k is a real number
not equal to 0
Pg 119 #1 – 6, 8 optional
Pg 120 #10, 13, 16, 18, 20 mandatory
2.5 Solve
Inequalities
Using
Technology
How are a polynomial inequality and the
graph of a polynomial function related?
Vocab: inequality, solution as a range, sign of a
function,
Pg 129 #1 – 5, optional
Pg 130 #6f, 7f, 8f, 9f, 11, 12, 14 mandatory
2.6 Solve
Factorable
Polynomial
Inequalities
Algebraically
How can I solve factorable inequalities
algebraically?
Vocab:
Pg 138 # 1 – 4, 11, 12 optional
Pg 139 #5de, 6cd, 7cd, 8, 9 mandatory
Skills: write an equation for a family of polynomial
functions with given characteristics (e.g. zeros, passing
through a point), sketch graphs of given families of
functions, expand and simplify equations involving whole
numbers, fractions and square roots, determine an equation
for a given graph of a polynomial, solve problems involving
factors of a polynomial
Skills: explain the relationship between the solution to an
equation and the solution to the corresponding inequality,
solve a polynomial inequality graphically and algebraically,
model a problem with a polynomial inequality and solve it
Skills: solve a linear inequality with non-zero terms on both
sides of the inequality sign, explain why it is necessary to
reverse the signs when multiplying or dividing with a
negative value, solve a polynomial inequality by factoring
and considering cases or using intervals in a table or with
number lines,
The Chapter will wrap up with a review, the Chapter 2 Task and a written test. Class discussions, class work, homework
and quizzes will help you to determine how well you understand the course material in preparation for the summative
assessments.
Extra help is available with me upon request.
There is free tutoring and instructional videos available on the class moodle.