Math 100
Section 1.2
Exponents and Radicals
Exercise 1: Evaluate each of the following expressions
a) b) (-5)2
c) -52d) (-5)3
e) (25)3f) 2322
In calculus, it is often useful to convert an expression of the form
into the form a-n. You should learn to easily switch expressions from one form to the other.
Exercise 2: Write the following expressions without negative exponents
a) a3b-5b) (x2 + 3x)-3
c) e-xd) (1 + 5x)-1
Exercise 3: Use negative exponents to eliminate the denominators from the following expressions.
a) b)
c) d)
Exercise 4: Evaluate each of the following expressions:
a) 23b)
c) (- 5)0d) -3.140
L a w s o f E x p o n e n t sLaw / Description / Examples showing why the law is true
1. aman =am + n / To multiply two powers of the same number, add the exponents
2. / To divide two powers of the same number, subtract the exponents.
3. (am)n = amn / To raise a power to a new power, multiply the exponents.
4. (ab)n = anbn / To raise a product to a power, raise each factor to the power.
5. / To raise a product to a power, raise both numerator and denominator to the power.
6. / To raise a fraction to a negative power, invert the fraction and change the sign of the exponent.
7. / A number raised to a negative exponent can be moved from numerator to denominator, or vice versa, if you cnage the sign of the exponent.
Exercise 5: Use the laws of exponents to simplify the following expressions:
a) b) (2x)3(3x)2
c) (5a2b)(3a-3b4)d) (2ab2)3
e) (-5c-1d-2)-2 f)
g) h)
A cautionary note:
We have seen that (ab)n = anbn. This allows us to eliminate the brackets around a product raised to a power simply by raising each of the things inside the bracket to the power. Note that a similar law does not apply to sums.
(a + b)n an + bn
Exercise 6:
a) Show that (32)2 = 3222
b) Show that (3 + 2)2 32 + 22
Radicals and Rational Exponents
An expression that uses the symbol: is said to be in radical notation. If the n is missing, the symbol denotes a positive square root: for example , since 42 = 16.
Exercise 5: Find each of the following, if possible:
a) b) c) -
d) e) f)
The left-hand side of the expressions in the above definitions is said to be in exponential notation, and the right-hand side is said to be in radical notation.
In pre-calculus and calculus, it is useful to be able to fluently switch from one notation to the other.
Exercise 7: Convert the expressions that are in exponential notation to radical notation, and vice versa.
a) b) c)
d) e) f)
g) h) i)
Exercise 8: The laws of exponents also hold for rational exponents. Use the law of exponents to simplify the following expressions as far as possible. (Convert from radical to exponential notation when it is convenient)
a) b)
c) d)
e) f)
Exercise 9:
a) Show that
b) Show that
Suggested Exercises and Summary of Objectives from Section 1.2
Suggested exercises are on page 24 of textbook.
1)To understand exponential notation including
a)Positive integer exponents ( #1a, 1b )
b)Negative and zero exponents ( #1c, 3a )
2)To learn and understand the laws of exponents, and use these laws to simplify algebraic expressions (# 15, 17, 19, 27, 31)
3)To understand radical expressions and rational exponents (#9, 33, 37, 47, 51, 55)
Math 100
Section 1.3
Algebraic Expressions
Adding and Subtracting Algebraic Expressions
When adding or subtracting algebraic expression, you may combine like terms. "Like terms" are terms with the same variables raised to the same powers.
Exercise 1: Find each sum or difference
a) (3x2 - x + 5) + (-8x2 + 3x - 9)
b) (x2 - 5x) - (3x2 - 4x - 1)
c) (a2 + 2ah + h2 + 3a + h - 5) - (a2 + 3a - 5)
Notice that terms are separated from other terms by plus (+) and minus (–) signs. If a plus or minus sign is tucked away inside a radical sign, then it doesn’t count: for our purposes, the radical must be treated as a monomial.
Monomials have 1 term, and no + or – signs (outside of a radical sign).
Binomials have 2 terms, and exactly one + or – signs (outside of a radical sign).
Trinomials have 3 terms, and exactly two + or - signs (outside of a radical sign).
Exercise 2: Identify each of the following as being either a monomial, a binomial, or a trinomial:
a) 3x2 b) 5xy4 – 15x – 25 c) 5xy4 – 15x
d) 5xy4 - 3 e) + h f) 5xy4 – 15x
Multiplying Algebraic Expressions
In Section 1.2, we multiplied monomials using the laws of exponents. Section 1.3 involves multiplying binomials and occasionally trinomials. Multiplying binomials or trinomials is quite a lot more work than multiplying monomials, because each term in one factor must be multiplied by each term in the other factor.
You may have seen the acronym FOIL used for multiplying binomials. It helps us remember that the product of two binomials is the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms.
Exercise 3: Fill in the blanks in the expression below
(A + B)(C + D) = ____ + ____ + ____ + ____
Exercise 4: Find the following products. Indicate whether you are multiplying binomials or monomials
1. a) (5y)2 b) (5 + y)2
2. a) (4x)(5x) b) (4 + x)(5 + x)
3. a) b)
Exercise 5: Perform the indicated operations and simplify
a) -3a3(6a2 - 5a + 2)
b) (a + h)2 - 3(a + h) - (a2 - 3a)
c)
d) (x + 3)(x - 3)
e)
f)
Factoring Algebraic Expression
"Factoring" is the reverse of multiplying:
(3)(2)(5) = 30 (x - 2)(x + 5) = x2 + 3x - 10
The things that are multiplied together are called the "factors". For example, (x - 2) and (x + 5) are the factors of x2 + 3x - 10.
The easiest type of factoring is "factoring out a common factor".
Exercise 6: Factor out the largest common factor possible in each of the following expressions
a) 2x2y + 6xy3b) 4a - 20ab
In general, when faced with an expression to be factored, we can classify it into one of several types:
- Difference of squares: A2 - B2 = (A + B)(A - B)
- Perfect square of a sum: A2 + 2AB + B2 = (A + B)2
- Perfect square of a difference: A2 - 2AB + B2 = (A - B)2
- Difference of cubes:A3 - B3 = (A - B)(A2 + AB + B2)
- Sum of cubes: A3 + B3 = (A + B)(A2 - AB + B2)
- Trinomial of FOIL type, such as: x2 - 8x + 15 = (x - 5)(x - 3)
Types 2 and 3 are just special cases of type 6. Usually we use a trial and error procedure to factor trinomials such as number 6.
Exercise 7: Factor the following expressions, each of which can be factored as a difference of squares. If necessary, factor out a common factor first.
a) x2 - 16b) 18a2b - 8b
c) d) 75z4y3 - 3y
Exercise 8: Factor the following expressions, each of which can be factored as the perfect square of a sum or difference. If necessary, factor out a common factor first.
a) x2 + 10x + 25b) 4a2 - 12ab + 9b2
c) 2xy2 - 28xy + 98xd) 3x2 + 24x + 48
Exercise 9: Factor the following expressions, each of which can be factored as the sum or difference of cubes. If necessary, factor out a common factor first.
a) 8x3 + y3b) 54y3 - 2
c) x6 + 1d) 64x3 - 27y6
Exercise 10: Factor the following trinomials by the trial and error method. Where possible, factor out a common factor first.
a) x2 - 4x - 5b) 3x3 - 30x2 + 63x
c) x2 + 7x - 30d) 4xy2 + 52xy + 88x
Another method of factoring is known as "factoring by grouping". Use this method when an expression that contains four terms is to be factored. Factor out a common factor from the first two terms, then from the second two terms. If a common binomial results in each case, the binomial can then be factored as a common factor:
Exercise 11: Factor the following expressions by grouping:
a) x2y2 - 3y2 - 5x2 + 15
b) ab2 - 7b + a2b - 7a
The following flow chart summarizes what you need to know about factoring:
Exercise 12: Factor the following expressions as far as possible:
a) 2x3 - 6xb) 3x2 - 12x + 12
c) x2y + 5xy + 6yd) 9x2 - 12xy + 4y2
e) 16y2 - 9f) z4 + 125z
g) x2 - 1/4h) x2 + x - 2
i) x3 + 5x2 - 9x - 45j) 27x3 + 8
k) 3x5 - 3 x2 l) x3 - 3x2 - x + 3
Suggested Exercises and Summary of Objectives for Section 1.3
Suggested exercises are on pages 34-35.
1)To become skilled at adding, subtracting, multiplying and factoring algebraic expressions, in particular:
a) Multiplying by monomials, adding and subtracting (# 1, 3, 5, 7)
b) Multiplying two binomials (# 13, 15 )
c) Multiplying algebraic expressions involving radicals (# 9, 11, 31 )
d) Factoring out a common factor (# 41, 43, 57 )
e) Factoring expressions of the form ax 2 + bx + c by trial and error
(# 47, 49, 51)
f) Factoring a difference of squares x 2- y 2 (# 53 )
g) Factoring perfect squares x2 + 2xy + y2 (# 65, 69 )
h) Factoring differences and sums of cubes x3 + y3, x3 – y3 (# 63, 73, 81)
Math 100
Section 1.4
Fractional Expression
A Fractional expressions is a ratio of two algebraic expressions. Fractional expressions represent division. That is, a/b means a b. We have already learned how to simplify fractional expressions that involve only monomials. In this section we will deal with rational expressions that involve binomials, trinomials, etc.
Below is an example of a rational expression that has a binomial on the top and a trinomial on the bottom. By the way, the top part of the fraction is called a numerator, and the bottom is called a denominator (remember the denominator is downstairs. Later in this course, and in Calculus (if you continue your studies) you shall see that the numerator and denominator play very different rolls, and it is important to distinguish between them:
Simplifying Fractional Expressions by Cancellation
Cancelling is the process of removing factors that are common to both the numerator and denominator, and thus reducing the fraction to lowest terms. Example:
The important thing to note is that we can only cancel common factors. Remember, factors are things that are being multiplied.
Exercise 1 In each of the following determine whether or not the “cancelling” is correct:
a) b) c) d)
Multiplying and Dividing Fractional Expression
To multiply fractions, just multiply numerators by numerators, and denominators by denominators:
To divide fractions, just invert the second fraction and then multiply:
In practice, it is easier to factor the numerators and denominators and cancel where ever possible instead of actually carrying out the multiplication.
Exercise 2 In each case, use factoring and cancellation prior to doing the multiplication to arrive at the answer in an easier manner than that shown.
a)
b)
As you can see from the previous exercise, it is much easier and faster to factor and cancel common factors at the outset, rather than carry out the multiplication first. Of course it is possible that you will be asked to multiply in a situation where no cancellation is possible, but this seldom happens.
Exercise 3 Carry out the following operations and give the answer to lowest terms:
a)
b)
c)
c)
Adding and Subtracting Fractional Expressions
When fractional expressions have the same denominator, we can add or subtract them by adding or subtracting the numerators and retaining a common denominator:
and
If the denominators are different, we must first find a common denominator.
Exercsie 4 Combine into a single fraction and reduce to lowest terms:
a)
b)
c)
Simplifying Compound Fractions
Compound fractions contain a fraction in the numerator and/or denominator. The important thing to remember with compound fractions is to (if necessary) first combine the numerator into a single fraction and (if necessary) the denominator into a single fraction, then invert the denominator and multiply it by the numerator:
Exercise 5 Simplify the following compound fractions:
a)
b)
c)
Suggested Exercises and Summary of Objectives for Section 1.4
Suggested exercises are on pages 42-43.
Objectives:
To understand and become skilled at:
- Cancellation within a fractional expression (#1, 3, 5)
- Multiplying fractional expressions (# 7, 9)
- Dividing fractional expressions (# 11, 13)
- Adding/subtracting fractional expressions (#17, 21, 31)
- Compound fractions (# 37, 43)
Math 100
Section 1.5
Equations
An equation is a statement indicating that two algebraic expressions are equal. A value for the variable that satisfies the equation is said to be a solution to the equation. To solve an equation means to find the solution(s). An equation is characterized by the fact that it contains an equal sign: = . This may seem obvious, but sometimes students try to "solve" an expression that is not an equation, or use techniques that work well for solving equations to attempt to simplify expressions that are not equations, resulting in an error.
Exercise 1 Which of the following are equations and which are not?
a) x2 - 5x + 6b) x2 - 5x + 6 = 0
c) d)
The equal sign separates an equation into a left side and right side. We can change the value of one side of an equation as long as we change the value of the other side accordingly. For example, in Exercise 1 c) we could multiply both sides of the equation by 2(y - 1), and thus clear the denominators. The equation would look different after this multiplication, but it would be equivalent to the original equation in that it would have the same solutions (provided y 1). However, if we multiplied the expression in Exercise 1 d) by 2(y - 1), we would change its value and it would not be equivalent to the original expression.
The following is a list of some of the operations that may be performed to both sides of an equation to produce an equivalent equation.
Add the same thing to both sides
Subtract the same thing from both sides
Multiply both sides by anything but zero
Divide both sides by anything but zero
Raise both sides to a power
Linear Equations
Linear equations are equations in which the variable is not raised to any power besides 1. Linear equations do not contain variables in denominators or under radical signs. They are the simplest type of equations to solve and may be solved by isolating x on one side. This is accomplished by adding, subtracting, multiplying or dividing both sides by the same thing.
Exercise 2 Solve the following linear equations and check your solutions.
a) 2x - 5 = 9b) 7 + 3x = 4(x - 1)
c) d)
Quadratic Equations
A quadratic equation contains an x2 term, but no x3, x4, etc. Every quadratic equation can be put into the standard form:
ax2 + bx + c = 0,
where a, b, and c are numbers, and a 0.
We cannot solve a quadratic equation by isolating the variable on one side, because the equation usually contains a "x" term as well as an "x2" term. Since the terms are not "like" they cannot be combined. Thus a quadratic equation requires a different approach than a linear equation. We will review three methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula.
Solving quadratic equations by factoring
If the left-hand side of a quadratic equation (in standard form) can be factored into two binomials, then we may solve it by making use of the zero-product property: AB = 0 if and only if A = 0 or B = 0.
Exercise 3 Solve the following quadratic equations by factoring and making use of the zero-product property. If necessary, be sure to put the quadratic equation in standard form (with zero on one side) before factoring:
a) x2 + 10x + 21 = 0b) y2 - 7y = -10
c) x2 - 2x - 63 = -28d) x2 - 6x = 0
e) (x - 8)(x + 2) = 24f) y2 = 3y + 18
g) 5x2 - 5x - 280 = 0h) x2 - 25 = 0
Solving Quadratic Equations by "Completing the Square"
If a quadratic equation is of the form:
(x + n)2 = m
we can solve it by first taking the square root of both sides of the equation, and introducing a symbol on one side:
(x + n) =
and then by subtracting n from both sides:
x = - n
Exercise 4 Solve each of the following equations by the method just illustrated:
a) (x - 3)2 = 25b) (x + 7)2 = 10
If the quadratic equation is not already in the form (x + n)2 = m, we can put it into that form by a process known as "completing the square". The following exercise explains the origin of the name of the process.
Exercise 5
a) Find the area of the darkly shaded parts of the figure to the right, in terms of x.
b) What is the area small square missing in the lower right hand corner?
c) "Complete the square" by adding the area of the small missing square to the answer to part (a). You should now have a trinomial that can be factored into (x + 3)2, the total area of the (x + 3) by (x + 3) square. If total area of the large square is 144, what is x?
Exercise 6 Fill in the blank with the appropriate number to make a "perfect square" trinomial. Write each in factored form:
a) x2 - 12x + _____b) y2 + 20y + _____
c) t2 - 7t + ______d) x2 + + _____
Exercise 7 Solve the quadratic equation x2 + 10x + 8 = 0 by completing the square. Follow the steps given:
a) Subtract the constant term (8) from both sides of the equation.
b) Add the appropriate number needed to "complete the square" to both sides of the equation.
c) Factor the left side into a perfect square, and proceed as in exercise 4 to find the solutions.
Exercise 8 Solve the following quadratic equations by completing the square:
a) x2 + 6x + 7 = 0
b) x2 - 10x + 5 = 0
c) x2 + 3x - 1 = 0
d) 2x2 + 12x = 12 (divide both sides by 2 first)
e) 3x2 + 2x = 5(divide both sides by 3 first)
Solving Quadratic Equations Using the Quadratic Formula
The quadratic formula is derived by solving the general quadratic equation:
ax2 + bx + c = 0
by the method of completing the square. In the following exercise, assume a, b, and c are numbers and a 0.
Exercise 9 Derive the quadratic formula by following the steps given:
a)Divide both sides of ax2 + bx + c = 0 by a. (note we can do this since a 0)
b)Subtract the constant term (the term with no x or x2) from both sides.
c)Add the appropriate number (in terms of a and b) to both sides to complete the square.
d) Factor the left-hand side into a perfect square and solve for x. Express the results as a single fraction.
The quadratic formula tells us that the solutions to the equation
ax2 + bx + c = 0, where a 0 are given by:
A quadratic equations can have 0, 1, or 2 solutions, depending on the value of b2 – 4ac:
- If b2 – 4ac > 0, there are 2 solutions
- If b2 – 4ac = 0, there is 1 solution
- If b2 – 4ac < 0, there are no solutions, since the square root of a negative number is not a real number.
Exercise 10 Solve the following quadratic equations by using the quadratic formula.
a)x2 + 10x + 21 = 0
b) 3x2 + 2x = 7
c) 9x2 – 12x + 4 = 0
d) x2 + 5x + 8 = 0
Solving Other Equations
Exercise 11 Solve the following equations, which can be changed into linear or quadratic equations by a variety of manipulations. Check your answers.
a)
b)
c) (solve for i)
d)
e)
Suggested Exercises and Summary of Objectives for Section 1.5
Suggested exercises are on pages56-57
Objectives:
To understand and become skilled at:
1)Solving linear equations (#5, 11)
2)Solving equations that reduce to linear (#13, 17)
3)Solving quadratic equations:
a)By factoring (#29, 31)
b)By completing the square (#33, 35)
c)With the quadratic formula (#39, 41)
4)Solving equations involving radicals (#57, 61)
5)Solving a formula for an indicated variable (#79, 81, 87)
Math 100
Section 1.7
Inequalities
An inequality is just like an equation, except that in the place of the equal sign is one of the symbols , , , or . The solutions to an inequality are all the values of the variable that make the inequality true. Typically, the solutions to an inequality are intervals of numbers, rather than individual numbers.
Linear Inequalities
Linear inequalities are the simplest type of inequality and maybe solved in a manner similar to linear equations: isolate x on one side of the inequality symbol. Just as for linear equation, we may do this by adding, subtracting, multiplying or dividing both sides of the inequality by the same number. There is only one difference: if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality.
Exercise 1 What happens to the interval x 2 if all the numbers in the interval are multiplied by –1?
If x is a number that satisfies the inequality: x 2, then what may we conclude about – x?
Exercise 2 Solve the following linear inequalities. Graph the solutions on the number line.
a)x + 6 5x – 6
b)3x – 3 + 2x 1 – 7x – 10
Non-Linear Inequalities
Non-linear inequalities are considerably more work to solve than linear inequalities. Although non-linear inequalities are covered in the textbook in section 1.7, we will delay their study until Chapter 3 because there is a close relationship between non-linear inequalities and the graphs of polynomials and rational functions.
Suggested Exercises and Summary of Objectives for Section 1.7
Suggested exercises are on pages 78-79
Objectives:
To understand and become skilled at solving linear inequalities (#7, 9, 15)
We will return to Section 1.7 later in the course when we study non-linear inequalities.
Math 100 Chapter 1 Review Material1