Chapter 12 Exponents and Polynomials. Martin-Gay, Developmental Mathematics 2 12.1 – Exponents 12.2 – Negative Exponents and Scientific Notation 12.3

Chapter 12

Exponents and Polynomials

Martin-Gay, Developmental Mathematics 2

12.1 – Exponents

12.2 – Negative Exponents and Scientific Notation

12.3 – Introduction to Polynomials

12.4 – Adding and Subtracting Polynomials

12.5 – Multiplying Polynomials

12.6 – Special Products

12.7 – Dividing Polynomials

Chapter Sections

§ 12.1

Exponents


Exponents

Exponents that are natural numbers are shorthand notation for repeating factors.

34 = 3 • 3 • 3 • 3

3 is the base

4 is the exponent (also called power)

Note by the order of operations that exponents are calculated before other operations.


Evaluate each of the following expressions.

34 = 3 • 3 • 3 • 3 = 81

(–5)2 = (– 5)(–5) = 25

–62 = – (6)(6) = –36

(2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512

3 • 42 = 3 • 4 • 4 = 48

Evaluating Exponential Expressions

Example


Evaluate each of the following expressions.

Evaluating Exponential Expressions

Example

a.) Find 3x2 when x = 5.

b.) Find –2x2 when x = –1.

3x2 = 3(5)2 = 3(5 · 5) = 3 · 25

–2x2 = –2(–1)2 = –2(–1)(–1) = –2(1)

= 75

= –2


Product Rule (applies to common bases only)

am · an = am+n

Simplify each of the following expressions.

32 · 34 = 36 = 3 · 3 · 3 · 3 · 3 · 3= 729

x4 · x5 = x4+5

z3 · z2 · z5 = z3+2+5

(3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6

= 32+4

= x9

= z10

The Product Rule

Example


Power Rule

(am)n = amn


(23)3 = 29 = 512

(x4)2 = x8

= 23·3

= x4·2

The Power Rule

Example


Power of a Product Rule(ab)n = an · bn

The Power of a Product Rule

Example

Simplify (5x2y)3.

= 53 · (x2)3 · y3 = 125x6 y3(5x2y)3


Power of a Quotient Rule

The Power of a Quotient Rule

Example

Simplify

= 53 · (x2)3 · y3 = 125x6 y3(5x2y)3

n n

na ab b

n n

na ab b


The Power of a Quotient Rule

Simplify the following expression.4

3

2

3

r

p 43

42

3r

p

434

42

3 r

p

(Power of product rule)

12

8

81r

p

(Power rule)

Power of a Quotient Rule

Example

n n

na ab b


The Quotient Rule

Simplify the following expression.

Quotient Rule (applies to common bases only)

Example

0

mm n

na aaa

2

74

3

9

ab

ba 533 ba))((3 2714 ba

2

74

3

9

b

b

a

a

Group common bases together


Zero exponent

a0 = 1, a 0

Note: 00 is undefined.


50 = 1

(xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1

–x0 = –(x0) = – 1

Zero Exponent

Example

§ 12.2

Negative Exponents and Scientific Notation


Negative Exponents

Using the quotient rule from section 3.1,

0

2646

4

x

xxx

x

But what does x -2 mean?

26

4 11

xxxxxxxxx

xxxx

x

x


So, in order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows.

If a 0, and n is an integer, then

nn

aa

1

Negative Exponents


Simplify by writing each of the following expressions with positive exponents or calculating.

23

1

9

1

7

1

x

4

2

x Remember that without parentheses, x

is the base for the exponent –4, not 2x

23

7x

42 x

Simplifying Expressions

Example


Simplify by writing each of the following expressions with positive exponents or calculating.


Example

2323

1

9

1

2)3( 2)3(

1

9

1 Notice the difference in results when the

parentheses are included around 3.

31x

3x


Simplify by writing each of the following expressions with positive exponents.

3

1x

1)

3

11

x

2) 4

2

y

x

4

2

1

1

y

x2

4

x

y

1

3x 3x

(Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from a numerator to a denominator, or vice versa, and switch the exponent to its positive value.)


Example


If m and n are integers and a and b are real numbers, then:

Product Rule for exponents am · an = am+n

Power Rule for exponents (am)n = amn

Power of a Product (ab)n = an · bn

Power of a Quotient 0,

bb

a

b

an

nn

Quotient Rule for exponents 0, aaa

a nmn

m

Zero exponent a0 = 1, a 0

Negative exponent 0,1

aa

an

n

Summary of Exponent Rules


Simplify by writing the following expression with positive exponents or calculating.

2

374

32

3

3

ba

ba 2374

232

3

3

ba

ba

Power of a quotient rule

232724

22322

3

3

ba

ba

Power of a product rule

6148

264

3

3

ba

ba

Power rule for exponents

62614843 ba

Quotient rule for exponents

4843 ba4

8

81b

a668

2144

3

3

ba

ba

Negative exponents

44

8

3 b

a

Negative exponents



In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers.

A positive number is written in scientific notation if it is written as a product of a number a, where 1 a < 10, and an integer power r of 10.

a 10r

Scientific Notation


To Write a Number in Scientific Notation

1) Move the decimal point in the original number to the left or right, so that the new number has a value between 1 and 10.

2) Count the number of decimal places the decimal point is moved in Step 1.

• If the original number is 10 or greater, the count is positive.

• If the original number is less than 1, the count is negative.

3) Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.

Scientific Notation


Write each of the following in scientific notation.

47001) Have to move the decimal 3 places to the left, so that the new number has a value between 1 and 10.

Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3.

4700 = 4.7 103

0.000472) Have to move the decimal 4 places to the right, so that the new number has a value between 1 and 10.

Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4.

0.00047 = 4.7 10-

4

Scientific Notation

Example


To Write a Scientific Notation Number in

Standard Form• Move the decimal point the same number of

spaces as the exponent on 10.• If the exponent is positive, move the

decimal point to the right. • If the exponent is negative, move the

decimal point to the left.

Scientific Notation


Write each of the following in standard notation. 5.2738 1031)

Since the exponent is a positive 3, we move the decimal 3 places to the right.

5.2738 103 = 5273.8

6.45 10-52)

Since the exponent is a negative 5, we move the decimal 5 places to the left.

00006.45 10-5 = 0.0000645

Scientific Notation

Example


Operations with Scientific Notation

Example

Multiplying and dividing with numbers written in scientific notation involves using properties of exponents.

Perform the following operations.

= (7.3 · 8.1) (10-2 · 105) = 59.13 103

= 59,130

(7.3 10-2)(8.1 105)

1)

2) 9

4

104

102.1

9

4

10

10

4

2.1 5103.0 000003.0

§ 12.3

Introduction to Polynomials


Polynomial Vocabulary

Term – a number or a product of a number and variables raised to powers

Coefficient – numerical factor of a term

Constant – term which is only a number

Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.


In the polynomial 7x5 + x2y2 – 4xy + 7There are 4 terms: 7x5, x2y2, -4xy and 7.

The coefficient of term 7x5 is 7,

of term x2y2 is 1,

of term –4xy is –4 and

of term 7 is 7.

7 is a constant term.

Polynomial Vocabulary


Monomial is a polynomial with 1 term.

Binomial is a polynomial with 2 terms.

Trinomial is a polynomial with 3 terms.

Types of Polynomials


Degree of a termTo find the degree, take the sum of the exponents on the variables contained in the term.

Degree of a constant is 0.

Degree of the term 5a4b3c is 8 (remember that c can be written as c1).

Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial.

Degree of 9x3 – 4x2 + 7 is 3.

Degrees


Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved.

Find the value of 2x3 – 3x + 4 when x = 2.

= 2( 2)3 – 3( 2) + 42x3 – 3x + 4

= 2( 8) + 6 + 4

= 6

Evaluating Polynomials

Example


Like terms are terms that contain exactly the same variables raised to exactly the same powers.

Combine like terms to simplify.

x2y + xy – y + 10x2y – 2y + xy

Only like terms can be combined through addition and subtraction.

Warning!

11x2y + 2xy – 3y= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y =

= x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together)

Combining Like Terms

Example

§ 12.4

Adding and Subtracting Polynomials


Adding PolynomialsCombine all the like terms.

Subtracting PolynomialsChange the signs of the terms of the polynomial being subtracted, and then combine all the like terms.



= 3a2 – 6a + 11

Add or subtract each of the following, as indicated.

1) (3x – 8) + (4x2 – 3x +3)

= 4x2 + 3x – 3x – 8 + 3

= 4x2 – 5

2) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8

3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)

= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7

= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7

= 3x – 8 + 4x2 – 3x + 3

Example



In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression.

You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically.


§ 12.5

Multiplying Polynomials


Multiplying polynomials• If all of the polynomials are monomials, use the

associative and commutative properties.• If any of the polynomials are not monomials,

use the distributive property before the associative and commutative properties. Then combine like terms.



Multiply each of the following.

1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3

2) (4x2)(3x2 – 2x + 5)

= (4x2)(3x2) – (4x2)(2x) + (4x2)(5) (Distributive property)

= 12x4 – 8x3 + 20x2 (Multiply the monomials)

3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5)= 14x2 + 10x – 28x – 20= 14x2 – 18x – 20


Example


Multiply (3x + 4)2

Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4).

(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4)

= 9x2 + 12x + 12x + 16

= 9x2 + 24x + 16


Example


Multiply (a + 2)(a3 – 3a2 + 7).

(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7)

= a4 – 3a3 + 7a + 2a3 – 6a2 + 14

= a4 – a3 – 6a2 + 7a + 14


Example


Multiply (3x – 7y)(7x + 2y)

(3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y)

= 21x2 + 6xy – 49xy + 14y2

= 21x2 – 43xy + 14y2


Example


Multiply (5x – 2z)2

(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z)

= 25x2 – 10xz – 10xz + 4z2

= 25x2 – 20xz + 4z2


Example


Multiply (2x2 + x – 1)(x2 + 3x + 4)

(2x2 + x – 1)(x2 + 3x + 4)

= (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4)

= 2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4

= 2x4 + 7x3 + 10x2 + x – 4


Example


You can also use a vertical format in arranging the polynomials to be multiplied.

In this case, as each term of one polynomial is multiplied by a term of the other polynomial, the partial products are aligned so that like terms are together.

This can make it easier to find and combine like terms.


§ 12.6

Special Products


The FOIL Method

When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method.

F – product of First terms

O – product of Outside terms

I – product of Inside terms

L – product of Last terms


= y2 – 8y – 48

Multiply (y – 12)(y + 4)

(y – 12)(y + 4)

(y – 12)(y + 4)

(y – 12)(y + 4)

(y – 12)(y + 4)

Product of First terms is y2

Product of Outside terms is 4y

Product of Inside terms is -12y

Product of Last terms is -48

(y – 12)(y + 4) = y2 + 4y – 12y – 48F O I L

Using the FOIL Method

Example


Multiply (2x – 4)(7x + 5)

(2x – 4)(7x + 5) =

= 14x2 + 10x – 28x – 20

F

2x(7x)F

+ 2x(5)O

– 4(7x)I

– 4(5)L

O

I

L

= 14x2 – 18x – 20We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product.

Using the FOIL Method

Example


In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products.

Squaring a Binomial

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

Multiplying the Sum and Difference of Two Terms

(a + b)(a – b) = a2 – b2

Special Products


Although you will arrive at the same results for the special products by using the techniques of this section or last section, memorizing these products can save you some time in multiplying polynomials.

Special Products

§ 12.7

Dividing Polynomials



Dividing a polynomial by a monomialDivide each term of the polynomial separately by the monomial.

a

aa

3

153612 3 aa

a

a

a

3

15

3

36

3

12 3

aa

5124 2

Example


Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide.



725643 7256431

4329

6

2585

37

8

6344

32

Divide 43 into 72.

Multiply 1 times 43.

Subtract 43 from 72.

Bring down 5.

Divide 43 into 295.



Bring down 6.

Divide 43 into 376.



Nothing to bring down.32168 .43

We then write our result as



As you can see from the previous example, there is a pattern in the long division technique.

DivideMultiplySubtractBring downThen repeat these steps until you can’t bring down or divide any longer.

We will incorporate this same repeated technique with dividing polynomials.



15232837 2 xxx

x4

xx 1228 2 35 x

5

1535 x

Divide 7x into 28x2.

Multiply 4x times 7x+3.

Subtract 28x2 + 12x from 28x2 – 23x.

Bring down – 15.

Divide 7x into –35x.

Multiply – 5 times 7x+3.

Subtract –35x–15 from –35x–15.

Nothing to bring down.

15

So our answer is 4x – 5.



86472 2 xxx

x2

xx 144 220 x

10

7020 x78

Divide 2x into 4x2.

Multiply 2x times 2x+7.

Subtract 4x2 + 14x from 4x2 – 6x.

Bring down 8.

Divide 2x into –20x.

Multiply -10 times 2x+7.

Subtract –20x–70 from –20x+8.

Nothing to bring down.

8

)72(

78x

x2 10We write our final answer as


Documents

Chapter 12 Exponents and Polynomials. Martin-Gay, Developmental Mathematics 2 12.1 – Exponents 12.2 – Negative Exponents and Scientific Notation 12.3