CHAPTER · PDF fileIntroduction to Polynomials . ... Algebraic expressions can involve...

CHAPTER

4 Polynomials: Operations

4.1 Integers as Exponents 4.2 Exponents and Scientific Notation 4.3 Introduction to Polynomials 4.4 Addition and Subtraction of Polynomials 4.5 Multiplication of Polynomials 4.6 Special Products 4.7 Operations with Polynomials in Several Variables 4.8 Division of Polynomials

OBJECTIVES

4.1 Integers as Exponents

a Tell the meaning of exponential notation. b Evaluate exponential expressions with exponents of 0

and 1. c Evaluate algebraic expressions containing exponents. d Use the product rule to multiply exponential

expressions with like bases. e Use the quotient rule to divide exponential expressions

with like bases.

OBJECTIVES

f Express an exponential expression involving negative exponents with positive exponents.

a Tell the meaning of exponential notation.

An exponent of 2 or greater tells how many times the base is used as a factor. For example,

In this case, the exponent is 4 and the base is a. An expression for a power is called exponential notation.

EXAMPLE

1 What is the meaning of 35? Of n4? Of (2n)3? Of 50x2? Of (–n)3? –n3?

We read exponential notation as follows: an is read the nth power of a, or simply a to the nth, or a to the n. We often read x2 as “x-squared.” The reason for this is that the area of a square of side x is x·x or x2.

We often read x3 as “x-cubed.” The reason for this is that the volume of a cube with length, width, and height x is x·x·x, or x3.

b Evaluate exponential expressions with exponents of 0 and 1.

Exponents of 0 and 1

EXAMPLE

b Evaluate exponential expressions with exponents of 0 and 1.

2 Evaluate 51, (–8)1, 30, (–7.3)0, (186,892,046)0

c Evaluate algebraic expressions containing exponents.

Algebraic expressions can involve exponential notation. For example, the following are algebraic expressions:

We evaluate algebraic expressions by replacing variables with numbers and following the rules for order of operations.

EXAMPLE

3 Evaluate 1000 – x4 when x = 5.

EXAMPLE

4 Area of a Compact Disk

The standard compact disc used for software and music has a radius of 6 cm. Find the area of such a CD (ignoring the hole in the middle).

EXAMPLE Solution

In Example 4, “cm2” means “square centimeters” and “≈” means “is approximately equal to.”

EXAMPLE

5 Evaluate (5x)3 when x = –2.

EXAMPLE Solution

d Use the product rule to multiply exponential expressions with like bases.

There are several rules for manipulating exponential notation to obtain equivalent expressions. We first consider multiplying powers with like bases:

The Product Rule

EXAMPLE

7 Multiply and simplify.

EXAMPLE

Multiply and simplify.

EXAMPLE Solution

e Use the quotient rule to divide exponential expressions with like bases.

The following suggests a rule for dividing powers with like bases, such as a5/a2:

Note that the exponent in a3 is the difference of those in a5 ÷ a2.

The Quotient Rule

For any nonzero number a and any positive integers m and n,

(When dividing with exponential notation, if the bases are the same, keep the base and subtract the exponent of the denominator from the exponent of the numerator.)

EXAMPLE

12 Divide and simplify.

EXAMPLE

Divide and simplify.

EXAMPLE Solution

Consider 53/57 and first simplify it using procedures we have learned for working with fractions:

Now apply the rule for dividing exponential expressions with the same bases:

It follows that:

Negative Exponent

EXAMPLE

16 Express using positive exponents. Then simplify.

EXAMPLE

Express using positive exponents. Then simplify.

EXAMPLE Solution

The rules for multiplying and dividing powers with like bases hold when exponents are 0 or negative.

EXAMPLE

Simplify. Write the result using positive exponents.

EXAMPLE Solution

Definitions and Rules for Exponents

CHAPTER

OBJECTIVES

4.2 Exponents and Scientific Notation

a Use the power rule to raise powers to powers. b Raise a product to a power and a quotient to a power. c Convert between scientific notation and decimal

notation. d Multiply and divide using scientific notation. e Solve applied problems using scientific notation.

a Use the power rule to raise powers to powers.

Consider an expression like (32)4. We are raising to the fourth power:

Note that in this case we could have multiplied the exponents:

The Power Rule

EXAMPLE

1 Simplify. Express the answers using positive exponents.

EXAMPLE

Simplify. Express the answers using positive exponents.

EXAMPLE Solution

When an expression inside parentheses is raised to a power, the inside expression is the base. Let’s compare 2a3 and (2a)3.

Raising a Product to a Power

EXAMPLE

b Raise a product to a power and a quotient to a power.

6 Simplify.

EXAMPLE

11 Simplify.

EXAMPLE Solution

Raising a Quotient to a Power

EXAMPLE

14 Simplify.

EXAMPLE

17 Simplify.

EXAMPLE Solution

c Convert between scientific notation and decimal notation.

Scientific notation makes use of exponential notation. Scientific notation is especially useful when calculations involve very large or very small numbers. The following are examples of scientific notation.

Niagara Falls: On the Canadian side, the amount of water that spills over the falls in 1 day during the summer is about 4.9793 × 1010 gal = 49,793,000,000 gal.

The mass of the earth: 6.615 × 1021 tons = 6,615,000,000,000,000,000,000 tons The mass of a hydrogen atom: 1.7 × 10–24 g = 0.0000000000000000000000017 g

Scientific Notation

Scientific notation for a number is an expression of the type

where n is an integer, M is greater than or equal to 1 and less than 10 (1 ≤ M < 10), and M is expressed in decimal notation. 10n is also considered to be scientific notation when M = 1.

You should try to make conversions to scientific notation mentally as much as possible. Here is a handy mental device. A positive exponent in scientific notation indicates a large number (greater than or equal to 10) and a negative exponent indicates a small number (between 0 and 1).

EXAMPLE

Convert to scientific notation.

EXAMPLE

Convert to scientific notation.

EXAMPLE

Convert mentally to decimal notation.

EXAMPLE Solution

d Multiply and divide using scientific notation.

Consider the product In scientific notation, this is

By applying the commutative and associative laws, we can find this product by multiplying 4 · 2 to get 8, and 102·103 = 105.

EXAMPLE

22 Multiply:

EXAMPLE Solution

EXAMPLE

23 Multiply:

EXAMPLE Solution

Consider the quotient . In scientific notation, this is

EXAMPLE

24 Divide:

EXAMPLE Solution

EXAMPLE

25 Divide:

EXAMPLE

25 Divide:

EXAMPLE

e Solve applied problems using scientific notation.

27 DNA

A strand of DNA (deoxyribonucleic acid) is about 150 cm long and cm wide. The length of a strand of DNA is how many times the width?

EXAMPLE Solution

e Solve applied problems using scientific notation.

CHAPTER

OBJECTIVES

4.3 Introduction to Polynomials

a Evaluate a polynomial for a given value of the variable. b Identify the terms of a polynomial. c Identify the like terms of a polynomial. d Identify the coefficients of a polynomial. e Collect the like terms of a polynomial. f Arrange a polynomial in descending order, or collect

the like terms and then arrange in descending order. g Identify the degree of each term of a polynomial and

the degree of the polynomial.

OBJECTIVES

h Identify the missing terms of a polynomial. i Classify a polynomial as a monomial, a binomial, a

trinomial, or none of these.

a Evaluate a polynomial for a given value of the variable.

Consider an expression like (32)4. We are raising to the fourth power:

Note that in this case we could have multiplied the exponents:

Monomial

A monomial is an expression of the type axn, where a is a real-number constant and n is a nonnegative integer.

Polynomial

A polynomial is a monomial or a combination of sums and/or differences of monomials.

The following algebraic expressions are not polynomials:

Expressions (1) and (3) are not polynomials because they represent quotients, not sums or differences. Expression (2) is not a polynomial because

and this is not a monomial because the exponent is negative.

When we replace the variable in a polynomial with a number, the polynomial then represents a number called a value of the polynomial. Finding that number, or value, is called evaluating the polynomial. We evaluate a polynomial using the rules for order of operations (Section 1.8).

EXAMPLE

1 Evaluate the polynomial when x = 2.

EXAMPLE

2 Evaluate the polynomial when x = 4.

EXAMPLE Solution

EXAMPLE

5 Medical Dosage

The concentration C, in parts per million, of a certain antibiotic in the bloodstream after t hours is given by the polynomial equation

Find the concentration after 2 hr.

To find the concentration after 2 hr, we evaluate the polynomial when t = 2.

EXAMPLE Solution

The polynomial equation in Example 5 can be graphed if we evaluate the polynomial for several values of t.

b Identify the terms of a polynomial.

Subtractions can be rewritten as additions. When a polynomial is written using only additions, the monomials being added are called terms.

EXAMPLE

8 Identify the terms of the polynomial

EXAMPLE

9 Identify the terms of the polynomial

If there are subtractions, you can think of them as additions without rewriting.

EXAMPLE Solution

c Identify the like terms of a polynomial.

When terms have the same variable and the same exponent power, we say that they are like terms.

d Identify the coefficients of a polynomial.

The coefficient of the term 5x3 is 5. In the following polynomial, the red numbers are the coefficients, 3, –2, 5, and 4:

EXAMPLE

12 Identify the coefficient of each term in the polynomial

EXAMPLE Solution

e Collect the like terms of a polynomial.

We can often simplify polynomials by collecting like terms, or combining like terms. To do this, we use the distributive laws. We factor out the variable expression and add or subtract the coefficients. We try to do this mentally as much as possible.

EXAMPLE

14 Collect like terms.

Note that using the distributive laws in this manner allows us to collect like terms by adding or subtracting the coefficients. Often the middle step is omitted and we add or subtract mentally, writing just the answer. In collecting like terms, we may get 0.

EXAMPLE

15 Collect like terms:

EXAMPLE Solution

EXAMPLE

Expressing a term like xn by showing 1 as a factor, 1 · xn, may make it easier to understand how to factor or collect like terms.

EXAMPLE Solution

f Arrange a polynomial in descending order, or collect the like terms and then arrange in descending order.

Note in the following polynomial that the exponents decrease from left to right. We say that the polynomial is arranged in descending order:

The associative and commutative laws allow us to arrange the terms of a polynomial in descending order.

EXAMPLE

20 Arrange the polynomial in descending order.

EXAMPLE Solution

EXAMPLE

21 Collect like terms and then arrange in descending order:

EXAMPLE Solution

We usually arrange polynomials in descending order, but not always. The opposite order is called ascending order. Generally, if an exercise is written in a certain order, we give the answer in that same order.

g Identify the degree of each term of a polynomial and the degree of the polynomial.

The degree of a term is the exponent of the variable. The degree of the term –5x3 is 3.

EXAMPLE

22 Identify the degree of each term of

The degree of a polynomial is the largest of the degrees of the terms, unless it is the polynomial 0. The polynomial 0 is a special case. We agree that it has no degree either as a term or as a polynomial. This is because we can express 0 as 0 = 0x5 = 0x7 and so on, using any exponent we wish.

Let’s summarize the terminology that we have learned, using the polynomial

h Identify the missing terms of a polynomial.

If a coefficient is 0, we generally do not write the term. We say that we have a missing term.

For certain skills or manipulations, we can write missing terms with zero coefficients or leave space.

EXAMPLE

h Identify the missing terms of a polynomial.

25 Write the polynomial in two ways: with its missing term and by leaving space for it.

i Classify a polynomial as a monomial, a binomial, a trinomial, or none of these.

Polynomials with just one term are called monomials. Polynomials with just two terms are called binomials. Those with just three terms are called trinomials. Those with more than three terms are generally not specified with a name.

i Classify a polynomial as a monomial, a binomial, a trinomial, or none of these.

CHAPTER

OBJECTIVES

4.4 Addition and Subtraction of Polynomials

a Add polynomials. b Simplify the opposite of a polynomial. c Subtract polynomials. d Use polynomials to represent perimeter and area.

EXAMPLE

a Add polynomials.

1 Add:

To add two polynomials, we can write a plus sign between them and then collect like terms.

EXAMPLE

a Add polynomials.

3 Add:

EXAMPLE Solution

a Add polynomials.

EXAMPLE

a Add polynomials.

4 Add:

b Simplify the opposite of a polynomial.

The opposite of x – 2x + 5 can be written as

We find an equivalent expression by changing the sign of every term:

Opposites of Polynomials

To find an equivalent polynomial for the opposite, or additive inverse, of a polynomial, change the sign of every term. This is the same as multiplying by –1.

EXAMPLE

5 Simplify:

EXAMPLE

6 Simplify:

EXAMPLE Solution

EXAMPLE

c Subtract polynomials.

8 Subtract:

Recall that we can subtract a real number by adding its opposite, or additive inverse: This allows us to subtract polynomials.

EXAMPLE Solution

EXAMPLE

10 Write in columns and subtract:

EXAMPLE

d Use polynomials to represent perimeter and area.

12 Find a polynomial for the sum of the areas of these rectangles.

Recall that the area of a rectangle is the product of the length and the width.

EXAMPLE Solution

EXAMPLE

13 Lawn Area

A water fountain with a 4-ft by 4-ft square base is placed in a park in a square grassy area that is x ft on a side. To determine the amount of grass seed needed for the lawn, find a polynomial for the grassy area.

EXAMPLE Solution

CHAPTER

OBJECTIVES

4.5 Multiplication of Polynomials

a Multiply monomials. b Multiply a monomial and any polynomial. c Multiply two binomials. d Multiply any two polynomials.

a Multiply monomials.

Consider (3x)(4x). We multiply as follows:

Multiplying Monomials

To find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents.

EXAMPLE

3 Multiply:

EXAMPLE Solution

After some practice, you will be able to multiply mentally.

b Multiply a monomial and any polynomial.

To find an equivalent expression for the product of a monomial, such as 2x and a binomial, such as 5x + 3, we use a distributive law and multiply each term of by 2x.

EXAMPLE

5 Multiply:

EXAMPLE Solution

Multiplying a Monomial and a Polynomial

To multiply a monomial and a polynomial, multiply each term of the polynomial by the monomial.

c Multiply two binomials.

To find an equivalent expression for the product of two binomials, we use the distributive laws more than once.

To visualize the product in the previous example, consider a rectangle of length x + 5 and width x + 4.

EXAMPLE

8 Multiply:

EXAMPLE Solution

d Multiply any two polynomials.

Consider the product of a binomial and a trinomial. We use a distributive law four times. You may see ways to skip some steps and do the work mentally.

Product of Two Polynomials

To multiply two polynomials P and Q, select one of the polynomials—say, P. Then multiply each term of P by every term of Q and collect like terms.

EXAMPLE

10 Multiply:

EXAMPLE

11 Multiply:

EXAMPLE Solution

EXAMPLE

12 Multiply:

When missing terms occur, it helps to leave spaces for them and align like terms as we multiply.

CHAPTER

OBJECTIVES

4.6 Special Products

a Multiply two binomials mentally using the FOIL method.

b Multiply the sum and the difference of two terms mentally.

c Square a binomial mentally. d Find special products when polynomial products are

mixed together.

Consider the product

This example illustrates a special technique for finding the product of two binomials:

To remember this method of multiplying, we use the initials FOIL.

The FOIL Method

To multiply two binomials, A + B and C + D, multiply the First terms AC, the Outside terms AD, the Inside terms BC, and then the Last terms BD. Then collect like terms, if possible.

EXAMPLE

1 Multiply:

EXAMPLE

10 Multiply:

EXAMPLE Solution

Consider the product of the sum and the difference of the same two terms, such as

Product of the Sum and Difference of Two Terms

The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term:

EXAMPLE

14 Multiply:

EXAMPLE Solution

c Square a binomial mentally.

Consider the square of a binomial, such as

Square of a Binomial

The square of a sum or a difference of two terms is the square of the first term, plus twice the product of the two terms, plus the square of the last term:

Multiplying Two Polynomials

EXAMPLE

d Find special products when polynomial products are mixed together.

22 Multiply:

EXAMPLE

d Find special products when polynomial products are mixed together.

22 Multiply:

CHAPTER

OBJECTIVES

4.7 Operations with Polynomials in Several Variables

a Evaluate a polynomial in several variables for given values of the variables.

b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial.

c Collect like terms of a polynomial. d Add polynomials. e Subtract polynomials. f Multiply polynomials.

A polynomial in several variables is an expression like those you have already seen, but with more than one variable.

EXAMPLE

1 Evaluate the polynomial when x = –2 and y = 5.

EXAMPLE Solution

The degree of a term is the sum of the exponents of the variables. The degree of a polynomial is the degree of the term of highest degree.

EXAMPLE

3 Identify the coefficient and the degree of each term and the degree of the polynomial

EXAMPLE Solution

b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial.

c Collect like terms of a polynomial.

Like terms have exactly the same variables with exactly the same exponents. For example, But and

EXAMPLE

c Collect like terms of a polynomial.

EXAMPLE

d Add polynomials.

7 Add:

We can find the sum of two polynomials in several variables by writing a plus sign between them and then collecting like terms.

EXAMPLE Solution

d Add polynomials.

EXAMPLE

e Subtract polynomials.

9 Subtract:

We subtract a polynomial by adding its opposite, or additive inverse.

EXAMPLE Solution

e Subtract polynomials.

EXAMPLE

f Multiply polynomials.

10 Multiply:

To multiply polynomials in several variables, we can multiply each term of one by every term of the other.

EXAMPLE Solution

EXAMPLE

17 Multiply:

Where appropriate, we use the special products that we have learned.

EXAMPLE Solution

CHAPTER

OBJECTIVES

4.8 Division of Polynomials

a Divide a polynomial by a monomial. b Divide a polynomial by a divisor that is a binomial.

a Divide a polynomial by a monomial.

When dividing a monomial by a monomial, we use the quotient rule of Section 4.1 to subtract exponents when the bases are the same. We also divide the coefficients.

To divide a polynomial by a monomial, we note that since it follows that

To divide a polynomial by a monomial, we divide each term by the monomial.

EXAMPLE

5 Divide:

EXAMPLE Solution

To check, we multiply the quotient by the divisor.

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

EXAMPLE

b Divide a polynomial by a divisor that is a binomial.

7 Divide by

Use long division when the divisor is not a monomial. Write polynomials in descending order and then write in missing terms.

EXAMPLE Solution

Now “bring down” the next term of the dividend—in this case, 6.

EXAMPLE Solution

The quotient is x + 3. The remainder is 0, expressed as R = 0. A remainder of 0 is generally not included in an answer.

EXAMPLE

8 Divide and check:

EXAMPLE Solution

If a remainder is not 0, continue dividing until the degree of the remainder is less than the degree of the divisor.

EXAMPLE

10 Divide and check:

EXAMPLE Solution

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