1 Associative, commutative, and distributive properties (Section p.1 , page 12)
2 Properties of opposites (Section P.1 , page 13)
3 Definition of subtraction (Section P.1 , page 13)
4 Rules for exponents (Section P.2 , page 27)
At one time, it was widely believed that heavy objects should fall faster than lighter objects, or, more exactly, that the speed of falling objects ought to be proportional to their weights. So if one object is four times as heavy as another, the heavier object should fall four times as fast as the lighter one. Legend has it that Galileo Galilei (1564–1642) disproved this theory by dropping two balls of equal size, one made of lead and the other of balsa wood, from the top of Italy’s Leaning Tower of Pisa. Both balls are said to have reached the ground at the same instant after they were released simultaneously from the top of the tower.
In fact, such experiments “work” properly only in a vacuum, where objects of different mass do fall at the same rate. In the absence of a vacuum, air resistance may greatly affect the rate at which different objects fall. Whether Galileo’s experiment was actually performed is unknown, but astronaut David R. Scott successfully performed a version of Galileo’s experiment (using a feather and a hammer) on the surface of the moon on August 2, 1971. It turns out that on Earth (ignoring air resistance), regardless of the weight of the object we hold at rest and then drop, the expression 16t2
In Example 1, we evaluate such expressions.
1 Learn polynomial vocabulary.
The height (in feet) of a golf ball above a driving range t seconds after being driven from the tee (at 70 feet per second) is given by the polynomial 70t−t2. After two seconds (t=2), the ball is 70(2)−(2)2=136 feet above the ground. Polynomials such as 70t−t2 appear frequently in applications.
We begin by reviewing the basic vocabulary of polynomials. A monomial is the simplest polynomial; it contains one term. In the variable x it has the form axk, where a is a constant and k is either a positive integer or zero. The constant a is called the coefficient of the monomial. For a≠0, the integer k is called the degree of the monomial. If a=0, the monomial is 0 and has no degree. Any variable may be used in place of x to form a monomial in that variable.
The following are examples of monomials.
2x5 | The coefficient is 2, and the degree is 5. |
−3x2 | The coefficient is −3, and the degree is 2. |
−7 | −7=−7(1)=−7x0. The coefficient is −7, and the degree is 0. |
8x | The coefficient is 8, and the degree is 1(x=x1). |
x12 | The coefficient is 1(x12=1⋅x12), and the degree is 12. |
−x4 | The coefficient is −1(−x4=(−1)⋅x4), and the degree is 4. |
The expression 5x−3 is not a monomial because the exponent on x is negative.
Two monomials in the same variable with the same degree can be combined into one monomial using the distributive property. For example, −3x5 and 9x5 can be added: −3x5+9x5=(−3+9)x5=6x5. Or −3x5 and 9x5 can be subtracted: −3x5−9x5=(−3−9)x5=−12x5. Two monomials in the same variable with the same degree are called like terms.
The symbols a0,a1,a2,…an in the general notation for a polynomial
are just constants; the numbers to the lower right of a are called subscripts. The notation a2 is read “a sub 2”, a1 is read “a sub 1”, and a0 is read “a sub 0”. This type of notation is used when a large or indefinite number of constants are required. The terms of the polynomial 5x2+3x+1 are 5x2, 3x, and 1; the coefficients are 5, 3, and 1; and its degree is 2.
Note that the terms in the general form for a polynomial are connected by the addition symbol +. How do we handle 5x2−3x+1? Because subtraction is defined in terms of addition, we rewrite
and see that the terms of 5x2−3x+1 are 5x2, −3x, and 1 and that the coefficients are 5, −3, and 1. (The degree is 2.) Once we recognize this fact, we no longer rewrite the polynomial with the + sign separating terms; we just remember that the “sign” is part of both the term and the coefficient. Polynomials and their properties will be discussed in more detail in Chapter 3.
Polynomials can be classified according to the number of terms they have. Polynomials with one term are called monomials, polynomials with two unlike terms are called bino-mials, and polynomials with three unlike terms are called trinomials. Polynomials with more than three unlike terms do not have special names.
By agreement, the only polynomial that has no degree is the zero polynomial, which results when all of the coefficients are 0. It is easiest to find the degree of a polynomial when it is written in descending order—that is, when the exponents decrease from left to right. In this case, the degree is the exponent of the leftmost term. A polynomial written in descending order is said to be in standard form.
In our introductory discussion about Galileo and free-falling objects, we mentioned two well-known polynomials.
16t2 gives the distance in feet a free-falling object falls in t seconds.
With v0=10, the expression 16t2+10t gives the distance an object falls in t seconds when it is thrown down with an initial velocity of 10 feet per second. Use these polynomials to
Find how far a wallet dropped from the 86th-floor observatory of the Empire State Building will fall in 5 seconds.
Find how far a quarter thrown down with an initial velocity of 10 feet per second from a hot air balloon will travel after 5 seconds.
The value of 16t2 for t=5 is 16(5)2=400. The wallet falls 400 feet.
The value of 16t2+10t for t=5 is 16(5)2+10(5)=450. The quarter travels 450 feet on its downward path after five seconds.
If 16t2+15t gives the distance an object falls in t seconds when it is thrown down at an initial velocity of 15 feet per second, find how far a quarter thrown down with an initial velocity of 15 feet per second from a hot air balloon has traveled after 7 seconds.
2 Add and subtract polynomials.
Like monomials, polynomials are added and subtracted by combining like terms. By convention, we write polynomials in standard form.
Find the sum of the polynomials
Horizontal Method: Group like terms and then combine them.
A second method for adding polynomials is to arrange them in columns so that like terms appear in the same column.
Column Method:
Find the sum of
Find the difference of the polynomials
Horizontal Method: First, using the definition of subtraction, change the sign of each term in the second polynomial. Then add the resulting polynomials by grouping like terms and combining them.
The second method for finding the difference of polynomials requires arranging them in columns so that like terms appear in the same column.
Column Method: To find the difference, we again change the sign of each term in the second polynomial and then add.
Find the difference of
3 Multiply polynomials.
Previously, we multiplied two monomials, such as −3x2 and 5x3, by using the product rule for exponents: am⋅an=am+n. We multiply a monomial and a polynomial by combining the product rule and the distributive property.
Multiply 4x3 and 3x2−5x+7.
Multiply −2x3 and 4x2+2x−5.
We can use the distributive property repeatedly to multiply any two polynomials. As with addition, there are two methods.
Multiply 4x2+3x and x2+2x−3.
Horizontal Method: We treat the second polynomial as a single term and use the distributive property, (a+b)c=ac+bc, to multiply it by each of the two terms in the first polynomial.
The column method resembles the multiplication of two positive integers written in column form.
Column Method:
Multiply 5x2+2x and −2x2+x−7.
The basic multiplication rule for polynomials can be stated as follows.
4 Use special-product formulas.
Particular polynomial products called special products occur frequently enough to deserve special attention. We introduce a very useful method, called FOIL, for multiplying two binomials. Notice that
Now consider the relationship between the terms of the factors in (x+2)(x−7) and the second line, x2−7x+2x−14, in the computation of the product.
Use the FOIL method to find the following products.
(2x+3)(x−1)=F(2x)(x)+O(2x)( −1)+I(3)(x)+L(3)(−1)=2x2−2x+3x−3=2x2+x−3Combine like terms.
(3x−5)(4x−6)=F(3x)(4x)+O(3x)(−6)+I(−5)(4x)+L(−5)(−6)=12x2−18x−20x+30=12x2−38x+30Combine like terms.
(x+a)(x+b)=Fx⋅x+Ox⋅b+Ia⋅x+La⋅b=x2+bx+ax+a⋅bxb=bx (commutative property)=x2+(b+a)x+abCombine like terms.
Use FOIL to find each product.
(4x−1)(x+7)
(3x−2)(2x−5)
We can find a general formula for the square of any binomial sum (A+B)2 by using FOIL.
A formula such as (A+B)2=A2+2AB+B2 can be used two ways.
Find (2x+3)2.
Pattern Method: In (2x+3)2, the first term is 2x and the second term is 3. Rewrite the formula as
Substitution Method: In the formula (A+B)2=A2+2AB+B2, substitute 2x=A and 3=B. Then
Find (3x+2)2.
The definition of subtraction allows us to write any difference A−B as the sum A+(−B). Because A−B=A+(−B), we can use the formula for the square of a binomial sum to find a formula for the square of a binomial difference (A−B)2.
We leave it for you to use FOIL to verify the formula for finding the product of the sum and difference of terms.
Find the product (3x+4)(3x−4).
We use the substitution method; substitute 3x=A and 4=B in the formula (A+B)(A−B)=A2−B2. Then
Find the product (1−2x)(1+2x).
The special-products formulas, such as the formulas for squaring a binomial sum or difference, are used often and should be memorized. However, you should be able to derive them from FOIL, if you forget them. We list several of these formulas next.
To this point, we have investigated polynomials only in a single variable. We next extend our previous vocabulary to discuss polynomials in more than one variable. Any product of a constant and two or more variables, each raised to whole-number powers, is a monomial in those variables. The constant is called the coefficient of the monomial, and the degree of the monomial is the sum of all the exponents appearing on its variables. For example, the monomial 5x3y4 is of degree 3+4=7 with coefficient 5.
Multiply.
(7a+5b)(7a−5b)
(2x+3y)3
(7a+5b)(7a−5b)=(7a)2−(5b)2(A+B)(A−B)=A2−B2=49a2−25b2Simplify.
(2x+3y)3=(2x)3+3(2x)2(3y)+3(2x)(3y)2+(3y)3(A+B)3=A3+3A2B+3AB2+B3=23x3+3(22x2)(3y)+3(2x)(32y2)+33y3Power-of-a-product rule=8x3+36x2y+54xy2+27y3Simplify.
Multiply.
(x+2y)(x−2y)
(2x−y)3
The polynomial −3x7+2x2−9x+4 has leading coefficient and degree .
When a polynomial is written so that the exponents in each term decrease from left to right, it is said to be in form.
When a polynomial in x of degree 3 is added to a polynomial in x of degree 4, the resulting polynomial has degree .
When a polynomial in x of degree 3 is multiplied by a polynomial in x of degree 4, the resulting polynomial has degree .
True or False. When FOIL is used, the terms 7 and x are called the inside terms of the product (4x+7)(x−2).
True or False. There are values of A and B for which (A+B)2=A2+B2.
True or False. The expression x3−3√x+7 is a polynomial of degree 3.
True or False. The expression x5−√2x3+11 is a polynomial of degree 5.
In Exercises 9–16, determine whether the given expression is a polynomial. If it is, write it in standard form.
1+x2+2x
x−1x
x−2+3x+5
3x4+x7+3x5−2x+1
x2−2|x|+4
x3+5√x+1
5x3−|π−7|x+11
2x2−√3x+4
In Exercises 17–20, find the degree and list the terms of the polynomial.
7x+3
−3x2+7
x2−x4+2x−9
x+2x3+9x7−21
In Exercises 21–30, perform the indicated operations. Write the resulting polynomial in standard form.
(x3+2x2−5x+3)+(−x3+2x−4)
(x3−3x+1)+(x3−x2+x−3)
(2x3−x2+x−5)−(x3−4x+3)
(−x3+2x−4)−(x3+3x2−7x+2)
(−2x4+3x2−7x)−(8x4+6x3−9x2−17)
3(x2−2x+2)+2(5x2−x+4)
−2(3x2+x+1)+6(−3x2−2x−2)
2(5x2−x+3)−4(3x2+7x+1)
(3y3−4y+2)+(2y+1)−(y3−y2+4)
(5y2+3y−1)−(y2−2y+3)+(2y2+y+5)
In Exercises 31–78, perform the indicated operations.
6x(2x+3)
7x(3x−4)
(x+1)(x2+2x+2)
(x−5)(2x2−3x+1)
(3x−2)(x2−x+1)
(2x+1)(x2−3x+4)
(x+1)(x+2)
(x+2)(x+3)
(3x+2)(3x+1)
(x+3)(2x+5)
(−4x+5)(x+3)
(−2x+1)(x−5)
(3x−2)(2x−1)
(x−1)(5x−3)
(2x−3a)(2x+5a)
(5x−2a)(x+5a)
(x+2)2−x2
(x−3)2−x2
(4x+1)2
(3x+2)2
(x+34)2
(x+25)2
(3x−4y)2
(2x+5y)2
(x2+2)2
(3−x2)2
(x2+2x)2
(3y−y3)2
(x+2)3
(2x−1)3
(x+3)3−x3
(x−2)3−x3
(x+2y)3
(2x+3y)3
(5+2x)(5−2x)
(3−4x)(3+4x)
(2x+3y)(2x−3y)
(5x−2y)(5x+2y)
(x+1x)(x−1x)
(y2−2y)(y2+2y)
(x2+3)(x2−3)
(x3−2)(x3+2)
(2x−3)(x2−3x+5)
(x−2)(x2−4x−3)
(1+y)(1−y+y2)
(y+4)(y2−4y+16)
(x−6)(x2+6x+36)
(x−1)(x2+x+1)
In Exercises 79–88, perform the indicated operations.
(x+2y)(3x+5y)
(2x+y)(7x+2y)
(2x−y)(3x+7y)
(x−3y)(2x+5y)
(x−y)2(x+y)2
(2x+y)2(2x−y)2
(x+y)(x−2y)2
(x−y)(x+2y)2
(x−2y)3(x+2y)
(2x+y)3(2x−y)
In Exercises 89–94, use a2+b2=(a+b)2−2ab=(a−b)2+2ab.
If x+y=4 and xy=3, find the value of x2+y2.
If x−y=3 and xy=2, find the value of x2+y2.
If x+1x=3, find the value of
x2+1x2.
x4+1x4.
If x−1x=2, find the value of
x2+1x2.
x4+1x4.
If 3x+2y=12 and xy=6, find the value of 9x2+4y2.
If 3x−7y=10 and xy=−1, find the value of 9x2+49y2.
Ticket prices. Theater ticket prices (in dollars) x years after 2006 are described by the polynomial −0.025x2+0.44x+4.28. Find the value of this polynomial for x=6 and state the result as the ticket price for a specific year.
Box-office grosses. Theater box-office grosses (in billions of dollars) x years after 2008 are described by the polynomial 0.035x2+0.15x+5.17. Find the value of this polynomial for 2012 and state the result as box-office gross for a specific year.
Paper shredding. A business that shreds paper products finds that it costs 0.1x2+x+50 dollars to serve x customers. What does it cost to serve 40 customers?
Gift baskets. A company will produce x (for x≥10) gift baskets of wine, meat, cheese, and crackers at a cost of (x−10)2+5x dollars per basket. What is the per basket cost for 15 baskets?
In Exercises 99 and 100, use the fact (from Example 1) that an object thrown down with an initial velocity of v0 feet per second will travel 16t2+v0t feet in t seconds.
Free fall. A sandwich is thrown from a helicopter with an initial downward velocity of 20 feet per second. How far has the sandwich fallen after five seconds?
Free fall. A ring is thrown off the Empire State Building with an initial downward velocity of 10 feet per second. How far has the ring fallen after two seconds?
Cruise ship revenue. A cruise ship decides to reduce ticket prices to its theater by x dollars from the current price of $22.50.
Write a polynomial that gives the new price after the x-dollar deduction.
The revenue is the number of tickets sold times the price of each ticket. If 30 tickets were sold when the price was $22.50 and 10 additional tickets are sold for each dollar the price is reduced (all tickets are sold at the same price), write a polynomial that gives the revenue in terms of x when the original price is reduced by x dollars.
Used-car rentals. A dealer who rents used cars wants to increase the monthly rent from the current $250 in n increases of $10.
Write a polynomial that gives the new price after n increases of $10.
The revenue is the number of cars rented times the monthly rent for each car. If 50 cars can be rented at $250 per month and 2 fewer cars can be rented for each $10 rent increase, write a polynomial that gives the monthly revenue in terms of n.
Show that (a+b+c)2=a2+b2+c2+2ab+2bc+2ca.
In Exercises 104–108 use Exercise 103.
Find (x2+x+1)2.
Simplify (2x+y−z)2−(2x−y+z)2.
If a+b+c=8 and ab+bc+ca=12, find the value of a2+b2+c2.
If x+y+z=12 and x2+y2+z2=44, find the value of xy+yz+zx.
If x2+y2+z2=64 and xy+yz+zx=18, find the value of x+y+z.
Show that (a+b+c)(a2+b2+c2−ab−bc−ca)=a3+b3+c3−3abc.
Use Exercise 109 to show that if a+b+c=0, then a3+b3+c3=3abc.
In Exercises 111–114 use Exercises 103 and 110 where applicable.
Show that (x−y)3+(y−z)3+(z−x)3=3(x−y)(y−z)(z−x).
Show that (2x−3y)3+(3y−5z)3+(5z−2x)3=3(2x−3y)(3y−5z)(5z−2x).
If a+b+c=8 and ab+bc+ca=19, find the value of a3+b3+c3−3abc.
If a+b+c=9 and ab+bc+ca=11, find the value of a3+b3+c3−3abc.
In Exercises 115–124 find the missing integers in each row. In Exercises 119–124 choose a and b with |a|<|b|.
a | b | a+b | ab | |
---|---|---|---|---|
115. | 3 | 4 | ||
116. | −3 | 5 | ||
117. | 2 | 6 | 8 | |
118. | −5 | −2 | −15 | |
119. | 7 | 10 | ||
120. | 8 | 15 | ||
121. | 2 | −35 | ||
122. | −2 | −35 | ||
123. | −5 | 6 | ||
124. | 5 | 6 |