Math 30-1
Polynomials, Radicals and
Rational Functions
Specific Outcome: Students will demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree with integral coefficients).
Achievement Indicators:
Explain how long division of a polynomial expression by a binomial expression of the form , is related to synthetic division.
Divide a polynomial expression by a binomial expression of the form , using long division or synthetic division.
Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function.
Explain the relationship between the remainder when a polynomial expression is divided by , and the value of the polynomial expression at (remainder theorem).
Explain and apply the factor theorem to express a polynomial expression as a product of factors.
Specific Outcome: Students will graph and analyze polynomial functions (limited to polynomial functions of degree .
Achievement Indicators:
Identify the polynomial functions in a set of functions, and explain the reasoning
Explain the role of the constant term and the leading coefficient in the equation of a polynomial function with respect to the graph of the function.
Generalize rules for graphing polynomial functions of odd or even degree.
Explain the relationship between:
the zeros of a polynomial function
the roots of the corresponding polynomial equation
the x-intercepts of the graph of the polynomial function.
Explain how the multiplicity of a zero of a polynomial function affects the graph.
Sketch, with or without technology, the graph of a polynomial function.
Specific Outcome: Students will graph and analyze radical functions (limited to functions involving one radical).
Achievement Indicators:
Sketch the graph of the function , using a table of values, and state the domain and range.
Sketch the graph of the function , given the graph of the function and explain the strategies used.
Compare the domain and range of the function , to the domain and range of the function , and explain why the domains and ranges may differ.
Describe the relationship between the roots of a radical equation and the x-intercepts of the graph of the corresponding radical function.
Determine, graphically, an approximate solution of a radical equation.
Specific Outcome: Students will graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials, or trinomials).
Achievement Indicators:
Graph, with or without technology, a rational function.
Analyze the graphs of a set of rational functions to identify common characteristics.
Explain the behavior of the graph of a rational function for the values of the variable near a non-permissible value.
Determine if the graph of a rational function will have an asymptote or a hole for the non-permissible value.
Match a set of rational functions to their graphs, and explain the reasoning.
Describe the relationship between the roots of a rational equation and the x-intercepts of the graph of the corresponding rational function.
Lesson 1: Long and Synthetic Division of Polynomials
Problem 1: Use long division to determine the quotient and remainder of . Verify the answer.
Problem 2: Examine the long-division statement:
After you divide, your answer can be written in two forms:
1.
2.
Problem 3: Find the quotient and remainder of . Then write the division statement.
Question: How could you check this answer?
For the statement, identify the value or expression that corresponds to:
The divisor
The dividend (x2
−4x−24¿
The quotient
Problem 4: Find the quotient and remainder when .
Synthetic Division
Use synthetic division to find Q(x) and R if . Explain the significance of the remainder.
Note: To use synthetic division, the divisor must be a binomial of degree 1.
Problem 5: Use synthetic division to find the quotient and remainder of each of the following.
(a) (b)
Lesson 2: The Remainder Theorem
5
INVESTIGATION!
Complete the table below. Identify the value of a in each binomial divisor of the form . Perform synthetic division to calculate the remainder, then substitute the value in to the polynomial and evaluate. Record these values in the last column of the table.
Polynomial Binomial Divisor
Value of a Remainder Result of substituting
into the polynomial
x3
+2x2−5x−6
Compare the values of each remainder from synthetic division to the value from substituting into . What do you notice?
Use your conjecture to predict the remainder when the polynomial 2x3
−4x2+3x−6 is divided by each binomial: Then verify your prediction using synthetic division.
(a)
(b)
Problem 1: Use the remainder theorem to find the remainder for the indicated division:
a)
b)
Problem 2: When is divided by the remainder is 25.
(a) Determinethe value of k.
(b) Determine the remainder when .
Problem 4: When is divided by the remainder is –27, and when is divided by the remainder is 103. Determine the values of a and b.
Assignment: Page 20 #4a)ii)iv),5,6,8,14
Problem 1:
(a) Divide by .
(b) Factor the Quotient, and write out the Division Statement.
(c) Use the Remainder Theorem to verify your remainder.
(d) What does this tell you about ?
The Factor Theorem:
For any polynomial, P(x),
If , then is a factor of P(x)
If , then is a factor of P(x)
If is a factor, then
If is a factor, then
Problem 2: Use the Factor Theorem to show that (x - 1) is a factor of the polynomial .
x a
a f
P aaf
0 x aProblem 3: Determine which of the following are factors of :
a) b)
c) d)
The Factor Property
If is a factor of the polynomial f(x), then ‘a’ must be a factor of the constant term of f(x).
Problem 5: Fully factor the polynomial
Assignment: Page 23 #10,11,12,MC#1 Page 31 #7
Lesson 4: Graphing Polynomial Functions
Define:
polynomial function: any function whose equation can be written in the form
degree: the value of the highest exponent
degree 1: linear function Example:
degree 2: quadratic function Example:
degree 3: cubic function Example:
degree 4: quartic function Example:
degree 5: polynomial function Example:
leading coefficient: the coefficient of the highest degree term ie.
y
=
4
x
2−
6
x
3−
4
constant term: the term independent of x.
Problem 1:Determine whether or not the following are polynomial functions. Justify your choices.
(a) ____________________________________________________________
(b) ____________________________________________________________________
(c) ________________________________________________________________
(d) __________________________________________________________________
Properties of Cubic Functions
Investigate: Using a graphing calculator, sketch
a) y=x
3 y
=2x3−7 x y= 1
2 x
3
−15 x−7
b)
y
= −
x
3y
=−
2
x
3+
7
x y
=−
5
(
x
+
2
) (
x
+
1
) (
x
−
1
)
In general: For cubic functions, and in fact any odd-degree function,…. i) if the leading coefficient is positive, a > 0, then
ii) if the leading coefficient is negative, a < 0, then
Problem 2: A cubic function is defined by .
a) Use technology to sketch the graph of this function.
b) Use the graphing calculator to determine the: i) y-intercept
ii) x-intercept(s)
iii) coordinates of any maximum and/or minimum points
Properties of Quartic Functions
Investigate: Using a graphing calculator, sketch
a)
y
=
x
4y
=
2
x
4−
x
3−
2
x
2+
1
b)
y
= −
x
4y
= −
2
x
4+
x
3+
11
x
2−
9
x
−
3
In general: For quartic functions, and in fact any even-degree function,…. i) if the leading coefficient is positive, a > 0, then
ii) if the leading coefficient is negative, a < 0, then
Problem 3: A quartic function is defined by .
a) Use technology to sketch the graph of this function.
b) Use the graphing calculator to determine the: i) y-intercept
ii) x-intercept(s)
iii) coordinates of any maximum and/or minimum points
iv) Domain and Range
Lesson 5: Relating Polynomial Functions and Equations
INVESTIGATION!
Relationship between roots, x-intercepts & zeros
Graph the function f(x)=x4+x3−10x2−4x+24 using graphing technology and determine the x-intercepts from the graph.
Factor f(x). Then use the factors to determine the zeros of f(x).
Remember when solving a polynomial functions you would set it equal to zero.
f(x)=x4+x3−10x2−4x+24=0.
What is the relationship between the zeros of a function, the x-intercepts of the corresponding graph, and the roots of the polynomial equation?
Relationship between zeros, x-intercepts and roots If P(x)a(xr1)(xr2)(xr3)...(xrn) then:
The zeros of are .
The x-intercepts of the graph of are at .
The roots of (solutions to) the equation are x= .
Multiplicity: the number of times the roots of the equation (or x-intercepts of the graph) occur.
If , then the graph of will have:
a crossing x-intercept at . This represents a root of multiplicity one.
a touching x-intercept at . This represents a root of multiplicity two. Touching points also occur with any even-degree factor.
a inflection x-intercept at . This represents a root of multiplicity three. Inflection points also occur with any odd-degree factor .
Problem 1: A quadratic function is defined by .
a) Find the x and y-intercepts and state their multiplicities.
b) State the value of the leading coefficient.
c) Determine the vertex and then sketch the graph of this function.
Problem 2: A cubic function is defined by .
a) State the x-intercepts and their multiplicities.
b) State the value of the leading coefficient.
c) Sketch the graph of this function.
Problem 3: A polynomial function is defined by .
a) State the x-intercepts and their multiplicities.
b) State the value of the leading coefficient.
Problem 4: A quartic function is defined by .
a) State the x-intercepts and their multiplicities.
b) State the value of the leading coefficient.
c) Sketch the graph of this function.
Problem 5: A cubic function has zeros of –1, –1, and 3, and passes through the point P(–2, 30).
a) Sketch the graph of this function.
b) Determine the equation of this function in factored form.
Problem 5: Give the minimum degree equation in factored form for each of the following graphs.
a)
Problem 6: Factor completely, then use this information to determine the intercepts of the polynomial function and sketch its graph.
Lesson 6: Modelling and Solving Problems with Polynomial
Functions
Problem 1:Bill is preparing to make an ice sculpture. He has a block of ice that is 3 ft wide, 4 ft high and 5 ft long. Bill wants to reduce the size of the block of ice to 24 by removing the same amount from each of the three dimensions.
a) Write a polynomial equation to model this situation.
Problem 2: A rectangular piece of cardboard measuring 10 cm long and 8 cm wide is made into an open box by cutting squares from the corners and turning up the sides.
a) Write an expression which relates the Volume, V, of the box to the side length, x, of each square cut out.
b) Use your graphing calculator, to graph V against x.
c) Algebraically and graphically determine what size of square should be cut to have a box with volume of 48cm3.
d) Graphically determine what size of square should be cut out to have a box with the maximum volume. Determine the dimensions of this box.
Lesson 7: Properties of Radical Functions
INVESTIGATION!
Consider the functions y = 2x + 1 & y=
√
2x+1 Describe in words what is happening mathematically with these two functions. y=2x+1 _________________________________________________________. y=
√
2x+1 ________________________________________________________. ____________________________________________________________________ ____________________________________________________________________ Graph the two functions, create a table of values and note any connections between the two graphs.
x y = 2x + 1 y=
√
2x+10 4 12 24
40
Five facts concerning square roots:
1. The square root of a negative number is undefined. 2. The square root of zero is zero.
3. The square root of a number is larger than the number when the number is between zero and one.
4. The square root of one is one.
Radical function: A radical function has the form , where is a function. The square root of a function is only defined for non-negative numbers, so the domain of
is the set of values of x for which
Value of f(x) f(x)<0 f(x)=0 0<f(x)<1 f(x)=1 f(x)>1
Relative Location of Graph of
y=
√
f(x)The graph of
y=
√
f(x) isundefined
The graphs of
y=
√
f(x) andy = f(x) intersect on the x-axis
The graph of
y=
√
f(x) isabove the graph of y = f(x)
The graph of
y=
√
f(x)intersects the graph of y = f(x)
The graph of
y=
√
f(x) isbelow the graph of y = f(x)
Problem 1: Graph and on the same grid.
State the domain and range of both.
Problem 2: The graph of is shown:
a) Sketch
b) Give the domain and range of and
Problem 3: State the coordinates of any invariant points when is transformed to .
Problem 4: The graph of is shown.
a) Sketch the graph of
b) State the invariant points.
c) State the domain and range of
Assignment: Page 89 Discuss the Ideas #1-3 Page 90 #5-8,11,MC#1,2
Lesson 8: Solving Radical Equations Graphically
INVESTIGATION!
Consider the radical equation
√
x−4=5 Identify any restrictions
Describe a graphical approach using technology. Is there another approach?
Use your graphical method to solve the equation.
Solve the equation algebraically
Problem 1: For the function :
a) Sketch the graph using technology. State the domain.
b) Determine the x-intercept using technology correct to 2 decimal places.
c) State the zero of the function.
d) Give the solution to the equation
Problem 2:Solve the equation by sketching the graph of the related function.
Assignment: Page 93 #9
Lesson 9: Graphing Rational Functions
Rational function: A rational function has the form , where and are polynomial functions and
Assignment: Page 104 #1-3
INVESTIGATION!
Lesson 10: Analyzing Rational Functions
Problem 1:For the function
f(x) x23x4
x4 :
a) Express the function in factored form.
b) Determine whether this function will have a vertical asymptote or a point of discontinuity or both. If it has a point of discontinuity, determine the coordinates.
c) Simplify the function.
d) Use the simplified function to sketch the graph.
If f(x)=
(x−a)(x−b)
x−c , then f(x) will have a vertical asymptote at x=c .
If f(x)=
(x−a)(x−b)
x−a , then the graph of f(x) will be identical to g(x)=x−b except
it will have a point of discontinuity (hole) at
(
a, g(a))
. If f(x)=
(x−a)(x−b)
(check your graph using technology)
Problem 2: For the function
f(x) x22x8
x25x4
:
a) Express the function in factored form.
b) State any non-permissible values of x and whether each indicates a hole or a vertical asymptote.
c) Simplify the function and state the equation of the horizontal asymptote.
d) State the domain and range.
e) Sketch the function.
Note: Consider f(x)= g(x) h(x)
If the degree of g(x)<¿ ¿ degree of h(x) , then f(x) will have a horizontal asymptote at y=0 .
If the degree of degree of h(x) , then f(x) will have no horizontal asymptote .
Problem 3: Sketch the graph and determine the domain, range, x and y intercepts, points of discontinuity, and the equations of the asymptotes for each of the following.
a)
y 3x x2 2x8
b)
Problem 4: Compare the functions and and explain any differences.
Problem 5: Match each equation of each rational function with the most appropriate graph. Explain your reasoning.
Assignment: Page 114 #4-6,9,10,12b,MC#1,2 Page 134 #3,4,6,MC#1,2
Graph B Graph C
Lesson 11: Solving Rational Equations Graphically
Problem 1:
a) Solve the equation x+
6
x+2−5=0 graphically.
b) Check your solution by solving the equation algebraically.
Recall:
The zeros of a function are equal to the ______________ of the corresponding graph of , which is also equal to the ___________ of the equation
We can solve a rational equation graphically by using one of 2 methods:
1. X-intercept Method: Express the equation equal to zero. Then enter the equation in your calculator and look for the x-intercept(s).
2. Intersection Method: Enter the left side of the equation in and the right side of the
Problem 2:Solve the equation
x+3
2x−6=2x− x
x−3 using 2 different graphical methods.
Problem 3:Confirm the results of Problem 2 by using algebra.
