Unit A: Introduction

Have the students reflect upon what steps are required. Determine the greatest common factor of the numerical coefficients. Determine what variables are common to both terms. Select those variables together with the smaller exponent found on the common variables. Divide, remembering to subtract the exponents. Check by multiplication.

Foundational Objectives

• To use interval notation to specify sets of numbers in multiple settings. Supported by learning objectives 1, 3, 4, and 5.

• To choose and use appropriate factoring strategies. Supported by learning objectives 2, 3, 4, and 5.

• To demonstrate an understanding of the conditions that cause a quotient of expressions to be zero, undefined, or indeterminate. Supported by learning objective 3.

• To demonstrate the ability to solve multiple types of inequalities. Supported by learning objectives 4 and 5.

• To demonstrate the ability to simplify absolute value expressions, solve absolute value equations and graph absolute value functions supported by learning objective 5.

• To verify solutions to equations and inequalities with and without the support of a graphing utility. Supported by learning objectives 4 and 5.

In the table above, it is assumed that

. The symbols

and

refer to positive and negative infinity. They do not denote real numbers but are rather used to indicate that the interval extends infinitely far in the positive or negative direction. The interval

is said to be an open interval. The endpoints of an open interval are excluded, hence the round brackets. The interval

is said to be a closed interval. The endpoints of a closed interval are included, hence the square brackets. A consequence of an open interval is that any point in such an interval is between two other points in that interval. The same is not true for closed intervals.

In interval notation, the first number (symbol) recorded is always smaller than the second. Thus

, and

are incorrectly written. Students should be reminded that the union of two sets A and B , denoted symbolically as

, consists of all of the elements that are in A or B (or both A and B ). The intersection of two sets A and B , denoted symbolically as

, consists of those elements that are in both A and B . These set operations will be used in the exercises related to interval notation. The statement

is written in interval notation as

. Mixing inequality symbols in a statement such as

is not appropriate.

lends itself more readily to the form

and this is preferred over the form

Write each of the following intervals using set notation and inequalities as illustrated in 1 to 7 above.

Use interval notation to describe the results of the following unions/intersections.

Convey the information in each statement using both an inequality and an interval.

1. If you want to be a pilot, your height, x , must be between 167 cm and 185 cm inclusive.

2. If your age, x , is less than 16 years, you cannot obtain a driver's license.

3. Water will be in a liquid state if the temperature is more than

and less than

Provide opportunities for the students to orally read inequalities that are represented in interval notation, set notation, and graphical form. Have the students provide examples of numbers that do and do not belong to the interval. (CCT)

Infinity and the real numbers are two important concepts that are frequently used and depended upon in Calculus. For this reason, students should be engaged in ongoing discussion of these two concepts to continue to develop understanding of them. (COM)

In order to prepare for piecewise functions, students could also be asked to draw the graph of simple functions given a particular domain or range (NUM).

To factor the following over a given set: a difference of squares, a sum or difference of cubes, expressions containing rational coefficients, expressions containing negative exponents, expressions of the form

( n a positive integer), quadratic and higher order polynomials.

Factoring is required in many different areas of Calculus such as finding the domain of a function, evaluating limits, determining continuity, graphical analysis, and problem solving. This objective is intended to review and extend the students' understanding and skill with factoring. Because of the emphasis on solving over different number sets, students will also need to be familiar with irrational, rational, real, and complex number sets.

It is important to have students multiply the factors above to determine the unique products. Review what it means to factor an expression over a particular set. Have the students factor expressions such as

and

over the set of rationals. Have the students discuss how

is a difference of squares with irrational (real) factors of

but no rational factors. Students should also recognize that

also has irrational (real) factors of

but no rational factors.

Just how far is far enough? Generally when factoring

over the set of reals, we stop at

. We do not then factor

. The process could never end.

In the same way, the students should explore factoring sums and differences of cubes such as

and

. Applying previous knowledge, the students should be able to determine that the binomial factors are irrational (real).

After initially factoring a difference of cubes as the product of a binomial and a trinomial, students will often try to then factor the trinomial. It is important to recall that a quadratic will factor over the set of rationals only if the discriminant is a perfect square. Have the students recall that if the discriminant is positive, but not a perfect square, then the trinomial can be factored over the set of irrationals, and if the discriminant is negative, it is possible to factor the trinomial over the set of complex numbers. Consider the difference of cubes

where s is a real nonzero number. Now

. Can the trinomial

ever be factored? Examine the value of the discriminant. In this case, it is

, which is clearly negative. Thus, the trinomial factor in the factors of a difference of cubes cannot be factored over the reals. The same holds true for the trinomial factor in the factors of a sum of cubes.

By removing fractional common factors, leaving the polynomial with only integer coefficients, a factorable expression may become apparent. Frequently fractional coefficients are removed from expressions before factoring.

Engage students in a sequence of factoring exercises, similar to the following, so that students will be able to discover how to remove common factors containing negative exponents.

Students already know that

and

. Have different groups of students find each of the following products.

Based on their observations, ask the students to write the factors of

. Have the students then generalize the above observations to write the factors of

Show students that expressions such as

can sometimes be factored by changing all terms to a common denominator and then removing this denominator as a common factor. Thus

Review the process of factoring polynomials of degree greater than 2 by grouping and by the factor theorem.

Factor each of the following as a difference of squares over the set of real numbers.

Factor each of the following as a sum or difference of cubes over the set of real numbers.

Factor each of the following completely over the set of rationals by initially regarding each expression as a difference of squares.

Factor each of the following completely over the set of rationals by initially regarding each expression as a difference of cubes.

By examining the discriminant, determine which of the following trinomials can be factored over the set of rationals.

Factor each of the following expressions so that the second factor contains no negative exponents and no fractional coefficients.

Factor each of the following polynomials over the rationals by using the factor theorem and division.

Have the students recall the relationship between factors of expressions and the zeros of relations (roots of equations) defined by the expression. Building upon this idea, have the students consider the implications of factors that exist over the rational number and real numbers on the values on the graphs of the relation.

This idea can be extended by having the students investigate factoring quadratic expressions such as

over the set of real numbers by solving

. Using the quadratic formula, the roots are

and

or factors of

. (This type of simplification is one that students should be encouraged to play around with and become comfortable doing as it is often done with derivatives to make simplification possible and analysis easier).

Give the students an opportunity to try factoring expressions in different ways. For example,

could be factored using the difference of squares to get

or by using the factor theorem to get

The students need to determine that the two factored forms are equivalent, but that using the difference of squares to find the factors gave a more complete factorization of the expression. As students try out different factoring strategies and compare results, students will develop a sense of what type of factoring to use in different expressions and why.

Students should be encouraged to consider the mathematical operations within expressions to determine the types of values that those expressions can take on, and how that relates to the existence or non-existence of factors. It is often easier to tell if an expression, such as

is always positive than to start by trying to solve that expression equal to zero to find a root and thus a factor. Students should explore expressions like

and

for different natural numbered values of n to determine the nature of the expressions (always positive, always negative, sometimes zero) to help develop overall understanding and comfort with expressions. The development of this type of understanding has implications for many calculus concepts.

To discuss ideas using own language and then later incorporating the related mathematical terminology. (COM)

To interact co-operate, and collaborate through classroom activities and initiatives. (PSVS)

As a discussion starter, students could be divided into small groups and each group asked to determine the “answers” to the following three questions:

Have the students reach consensus within their groups and have the groups report their results (COM, PSVS).

Students need to understand that division is meaningful if a quotient yields one and only one result that checks . Thus

has meaning because 5 is the only value, that if multiplied by 2, results in 10. Thus

, because 0 is the only number, that if multiplied by 10, gives 0. This can then be generalized to yield

and

. Students should be able to verbalize that a quotient will have a value of 0 if the numerator is 0 and the denominator is nonzero. Later in the course, it will be important to recognize that the graph of

will have an x -intercept at values of x for which

and

We cannot find a value k such that

, because no matter what value is chosen for k ,

and not 10. We say the result is undefined . There is a bit of a play on words here – if you cannot find an answer, then the quotient is undefined. Have the students create a table of values wherein the results of dividing 10 by positive (and also negative) numbers that approach 0 are shown. From Math B30, students should be able to sketch the graph of

and see the nature of the graph as x approaches 0 from either direction. There will be many opportunities later in the course to revisit this idea. Conclude with the result that

is undefined if

and

. Students should be able to verbalize that a quotient is undefined whenever the denominator is zero and the numerator is not zero. Later in the course, it will be important to recognize that the graph of

will have a vertical asymptote at values of x for which

and

Every value of k satisfies the statement

. That is, it could be argued that

, and so on. Since there is not a unique value for the quotient

, the result is meaningless (not having one answer) and we say that the result is indeterminate . Later in the course, it will be important to recognize that the graph of

will have a hole at values of x for which

The exercises related to this topic are an excellent context in which students can practise their understanding of factoring.

Example: Find the value(s) of x , if any, for which the function

is (a) zero, (b) undefined, (c) indeterminate. Explain your work.

Solution: Determine the zeros of both the numerator and the denominator. Factoring is often helpful.

Let N be the set of zeros for the numerator. Then

. Recall, from section A.2, that the quadratic factor of a sum or difference of cubes has no zeros.

Find the value(s) of x , if any, for which each function is (a) zero, (b) undefined, (c) indeterminate.

Students could be asked to determine the error in reasoning in the following “proof”.

Students could be asked to discuss the conditions under which an expression containing two variables, such as

, is zero, undefined, or indeterminate.

Just like infinity and the real numbers, 0 plays a large role in Calculus and in mathematics, in general. Engage the students in discussions of the role of zero and its use in expressions. Some examples are:

Students need to be able to solve inequalities so that later in the course students will be able to determine the intervals in which a function is increasing, decreasing, concave up, or concave down.

Students have solved linear inequalities in Math 9, polynomial inequalities in Math B30, and possibly some double inequalities in Math C30 (locating the start and end of the period for a trig function). This section is intended to broaden the student's ability to solve inequalities of many different types. Students will have the opportunity to practise their factoring and interval notation skills.

Students use many of the following inequality properties intuitively. Students have not, however, formally studied all of these properties.

Let a , b , c , d , and k be real numbers. Note that similar properties are obtained if < is replaced by

and > is replaced by

. Likewise similar properties are obtained if every < (

) is replaced by > (

) and every > (

) is replaced by < (

Students should be asked to solve single or double linear inequalities in one variable. Encourage students to complete the solution by writing the variable on the left side of the inequality, such as

. Insist that students express their solution using a combination of set and interval notation, for example

Students have likely had little experience in solving double linear inequalities or in working with the reciprocal inequality property. The double linear inequality is a good setting in which to practise using this property. For example, if

then by the reciprocal property

. Multiplying by 2 leads to the solution.

Students need to be careful to always exclude any values of x from their solution set that result in an expression being undefined.

To solve factorable quadratic inequalities, a sign analysis of factors can be used as shown below.

Draw a number line and identify the value of x that makes each factor 0. Determine the sign nature of each factor.

Above a number line, use

and

signs to indicate the nature of each factor to the left and right of the zero value for each factor as shown above. Then consider the sign nature of the product of the factors and indicate below the number line.

Students are quick to believe that if

, then

. Have students rewrite the inequality as

, factor and solve using a number line. Have the students discuss the results and what was forgotten in the original solution.

Ask the students how they might solve quadratic inequalities that are not easily factorable such as

. By considering the graph of

, students should recognize that this is a parabola opening upwards. The x -intercepts are easily determined using the quadratic formula. Thus the solution set can be determined to be

In addition, students should be asked to solve quadratic inequalities that have no x -intercepts, such as

, yielding solution sets of

and

respectively.

These results can also be confirmed by using decimal approximations on the number line and carrying out the sign test in each resulting interval.

Review the solving of polynomial inequalities of higher degree (Math B30) such as

by applying the factor theorem.

Have students solve inequalities that contain quotients of functions using a sign analysis of factors/expressions. Students must be careful to exclude values of the variable that cause the quotient to become undefined or indeterminate. Students should always express their inequality in terms of 0, for example

before doing the sign analysis. Multiplying each term of an inequality by a variable expression is discouraged unless the sign nature of the variable expression is known.

Solve each of the following inequalities. Specify the solution set using set and interval notation.

(Use the reciprocal property because you cannot “multiply through” by x , since x may be positive, zero, or negative. If students multiply through by x , they must consider three separate cases:

. It is much simpler to use the reciprocal property.)

10. The length of a rectangle is 10 cm more than twice the width. If the perimeter of the rectangle is less than 62 cm, find the possible dimensions of the width.

11. To rent a car for one day, a company charges $50 for insurance plus $.25 per kilometre. How far did Ben travel if his rental charge was less than $85?

Solve each of the following inequalities. Express your answer using set and interval notation.

(Dividing by x is not workable as you will lose part of your solution set. In addition, if you divided by x , would you reverse the inequality sign? Of course that depends on whether x is positive or negative. Could you divide by x if x was zero? There are far too many questions to consider. Express the inequality as

, factor, and analyze the sign nature of each factor.)

15.

(Write with a common denominator,

, and determine the sign nature of

, x , and

16.

(Be sure to exclude

from the solution set as

makes the expression indeterminate. You will also need to recall that the quadratic factor of

, namely

, is always positive.

17. Two numbers have a sum of 10. If their product must be larger than

, find the possible values for each of the numbers.

Students should be encouraged to use a graphing calculator to support their solutions to inequalities. The solution to

could be obtained by having students graph

, and

. By using the calculator to determine the points of intersection of the graphs, the solution set can be obtained. Five of the significant screens that lead to the solution

are shown below.

By having the students represent the inequalities by functions students begin to relate the inequalities to the relationship between functions which is an important undertaking in differential and integral Calculus.

Students could also be challenged to solve more abstract inequalities such as: solve for x if a , b , and c are negative constants.

It is very important for students to think about what an expression tells them such as: always positive, always negative, when it is zero, when it is indeterminate. Moreover, connecting these ideas with the graphs and intervals of inequality provides the students with a sound platform for their upcoming Calculus studies.

Students can solve some inequalities using a written argument if the sign nature of the expression involved is clear. For example, the inequality

has solution

as the numerator is never negative and the denominator is always positive, the quotient could never be negative. The quotient could be zero, however, if

, so 4 is excluded from the solution set. The solution set can also be written as

Students should have an opportunity to solve fractional inequalities that contain trigonometric, logarithmic, and exponential functions. Some examples are:

(Careful, x , must be positive in order for

to be defined. Students should analyze the sign nature of

and

To apply the definition of absolute value to rewrite absolute value expressions, to use operations with absolute value, to use properties of absolute value to solve absolute value equations and inequalities, and to draw graphs of functions containing absolute value.

To make and justify decisions based upon understanding of calculus concepts (CCT).

To represent problems and understandings through a variety of communication modes (COM).

To interact, co-operate, and collaborate through classroom activities and initiatives (PSVS).

Absolute value functions provide a rich context through which students can gain a more complete understanding of graphs of functions, left and right handed limits, continuity, points at which derivatives are undefined, and absolute and relative minima/maxima. In order to use absolute value in these contexts, students must have a complete understanding of the definition of absolute value together with the absolute value properties. Realistically, most students at this stage only remember that the absolute value of a number makes the number positive. In Math 20, students solved absolute value equations such as

. Students have not solved absolute value inequalities, nor have they dealt with absolute value equations that contain more than one set of absolute value signs.

The development of the definition of absolute value could begin by asking the students to indicate if the expression, b , is positive, negative, or zero. Hopefully, their response would be that b could be any of these because, as a variable, it represents any number. Follow that by asking if

is positive, negative, or zero. Students often think that because you read

, as “negative b ”,

must be negative. Having just seen, however, that b can be positive, negative, or zero, students will generally have little trouble in realizing that

can also be positive, negative, or zero, depending on the value of b . This is a critical step.

Next refresh the students' understanding of absolute value by recalling the informal definition. The absolute value of the number b , denoted by

, is the distance that b is from zero on the number line. Since distance is never negative,

. Try some numerical calculations with the class including

and

Establish the formal definition of absolute value, namely

. This is a piecewise definition and student familiarity with it will also be beneficial in Unit B when the students explore piecewise functions.

Students need to be able to write absolute value expressions without the use of the absolute value sign. Based on the definition of absolute value, students should be able to write

as follows:

. By performing a sign analysis of an expression, students can easily determine those values of x for which the expression is positive and those values for which the expression is negative. Wherever the expression is negative, the absolute value of the expression is

Consider the task of rewriting

without absolute value signs. The completed sign analysis of

appears below.

Thus

. Note that the choice as to where to use square or round brackets is arbitrary as the absolute value of 0 is 0, and both

and

have a value of 0 for

and

. It is important that all values of x that make the expression defined are included in the rewriting of the absolute value expression.

As another example, have the students explain why

for all values of x . An analysis of the graph

shows it is a parabola that opens upwards and it has a negative discriminant, thus the graph has no x -intercepts and lies entirely above the x -axis. Therefore,

is always positive.

Note that

causes the expression to be undefined and is thus excluded in all pieces. Similarly

is excluded because it causes the expression to be indeterminate.

You could develop these by giving students a list of possible absolute value properties that include the ones above plus some statements that are not correct such as

and so on. Have students identify from this list of “properties” those that actually hold (CCT).

Students are of the impression that

, because throughout the development of radicals in Math 20 and Math A30, students are asked to assume that variables represent positive numbers. In Calculus, that assumption is not valid so it is critical that students understand that

and more generally that

if n is a positive even integer. Note that

if n is an odd positive integer.

To rewrite the expression

without the use of absolute value signs, the following approach, known as an expression analysis, is helpful.

The values of x at which the sign nature of each expression changes partition the number line into three regions, labelled I, II, and III. Thus in region I, namely

. In region II, namely

, and in region III, namely

. The boundary value, such as

, can be included in either region I or region II.

The following properties for absolute value expressions in inequities should be introduced by using numerical examples. In the table, k is a non-negative number. Note also that these properties are valid if

replaces < and

replaces >.

The name “Triangle Inequality” arises from the geometry result that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

Be sure students can read the statement

as “negative k is less than or equal to

AND

is less than or equal to k ”. The word AND is critical. It implies the intersection of two inequalities.

In solving inequalities using the absolute value inequality properties, some students find it easier to understand if the expression between the absolute value signs is replaced by a

. For example if students are presented with

, write a

in place of the

so that the absolute value inequality looks like

. This leads to

which then leads to

Have the students discuss how the absolute value of an expression changes any negative value into a positive value. Therefore, graphing functions containing absolute value is simply a matter of reflecting any portion of the graph that dips below the x -axis above the x -axis. This could be introduced by using the graphing calculator to draw the graphs of

followed by

. By examining the graph of

, students can see that y is negative for

. Thus

will convert each of these negative values into values that are positive. Hence the reflection shown below.

Point out the sharp turning point at

. This becomes a significant concept later in the course when students discuss if the function is differentiable at

and the conditions for a relative minimum at

Follow that example by showing the graph of

and asking the students to predict what the graph of

would look like.

In pairs, one student could sketch the graph of a function and ask their partner to sketch the graph of the absolute value of the function (COM, PSVS, CCT). Each partner explains thinking related to own graph.

Rewrite each of the following absolute value expressions without absolute value signs. Begin by performing a sign analysis of the expression. Write your solutions in the form

using interval notation.

Rewrite each of the following absolute value expressions without absolute value signs. Begin by performing a sign analysis of the expression. Using interval notation, write your solutions in the form

Use an absolute value property to explain why each of the following statements is true.

Rewrite each of the following radicals in simplest form. Note that variables represent any real numbers.

In the context of all real numbers, why is it permissible to write

but incorrect to write

Rewrite each of the following expressions without absolute value signs. Begin by performing an expression analysis. Support your solutions with interval notation.

Solve each of the following inequalities or equations. Specify your solution set.

Use the inequality property to write

. Now solve the double linear inequality.

Use the inequality property to write

. Solve each of these inequalities independently and find the union ( or ) of the two solution sets.

, the inequality becomes

, which has solution

. If

, the inequality becomes

. Multiplying by

yields

. Thus

. The solution set would be specified as

The absolute value expressions

and

separate the number line into three regions. In the region

becomes

which has no solution. In the region

becomes

which has solution

. Thus in the region

, the solution is

. In the region

becomes

which is true for all real numbers. Thus in the region

the solution is

. Uniting the two solutions, we arrive at

The inclusion of transendental numbers such as

and e in different situations at the beginning of the course will help the students develop a strong understanding of the role. Without familiarity, students think of

and e as variables rather than as numbers which can lead to more complex errors in understanding.

Similarly, the (-) sign should be included in a variety of roles, including as a power

. This is not a negative expression, but represents a positive rational expression.

Have the students compare and contrast

and

(CCT). Recognizing that some inverse operations result in a simplified statement involving an absolute value expression such as

is important for explorations of other functions such as natural logarithm.

Encourage the students to try different values to check properties or suggested properties (looking for a contradiction). Remind the students to consider all types of numbers as students frequently check only whole numbers.

Have the students explore other absolute value relationships such as if b and c are any two real numbers, then the distance between b and c on the number line is given by either

Regularly have the students reflect upon and discuss what they understand about expressions and their analysis (COM). Questions such as, what is the effect of squaring, what makes a fraction zero, and where might the denominator of an expression be problematic will help students develop a more intuitive and indepth understanding that will transfer to functions.

Have the students discuss how they understand and remember the absolute value inequality properties (COM). Asking students to, then, verify solutions to absolute value inequalities and equations using a graphing calculator is a worthwhile experience. For example the equation

is seen to have solutions

and

by drawing the graphs of

and

and determining the points of intersection. The significant windows are shown below.

Have the students explore and then prove that the maximum of two numbers, a and b , can be given by

Provide students a graph that is of some function

and ask the students to discuss what f(x) might look like (COM). This will help the students recognize that when an absolute value has been applied to an expression or function, some knowledge about that expression or function is lost. Students could discuss what else they would need to know in order to be able to sketch the true f(x).

Students could be asked to submit a proof for the triangle inequality property which is available in most Calculus texts (IL).

Graph each of the following functions without the aid of a graphing calculator. Then verify with a graphing calculator.