Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 1 § 5.2 More Work with Exponents and Scientific Notation

Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 1

§ 5.2

More Work with Exponents and

Scientific Notation


The Power Rule and Power of a Product or Quotient Rule for Exponents

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

The Power Rule

(ab)n = an · bn

n n

na ab b

Power Rule(am)n = amn

Power of a Product

Power of a Quotient


Simplify each of the following expressions.

(23)3 = 29 = 512

(x4)2 = x8

= 23·3

= x4·2

The Power Rule

Example:

= 53 · (x2)3 · y3 = 125x6 y3(5x2y)3

4

3

2

3

r

p 43

42

3r

p

434

42

3 r

p 12

8

81r

p


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

4

8

81b

a6 ( 14)4 8 2 63 a b

Quotient rule

Simplifying Expressions

Power rule

4 6 2

8 14 6

3

3a ba b

4 8 83 a b

Product rule

8

4 83ab

Negative exponents


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) (102 · 105) = 59.13 103

= 59,130

(7.3 102)(8.1 105)1)

2) 9

4

104

102.1

9

4

10

10

4

2.1 5103.0 000003.0


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

(7.3 102)(8.1 105)1)

2) 9

4

104

102.1

9

4

10

10

4

2.1 5103.0 000003.0

Operations with Scientific Notation

Example


§ 5.3

Polynomials and Polynomial Functions


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 one term.

Binomial is a polynomial with two terms.

Trinomial is a polynomial with three 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


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:


Adding PolynomialsTo add polynomials, combine all the like terms.

Adding Polynomials

Add.

(3x – 8) + (4x2 – 3x +3)

= 4x2 + 3x – 3x – 8 + 3

= 4x2 – 5

= 3x – 8 + 4x2 – 3x + 3

Example:


Subtracting PolynomialsTo subtract polynomials, add its opposite.

Subtracting Polynomials

Example:Subtract.

= 3a2 – 6a + 11

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

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

= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7

= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7


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.

Adding and Subtracting Polynomials


§ 5.4

Multiplying Polynomials


Multiplying Two PolynomialsTo multiply any two polynomials, use the distributive property and multiply each term of one polynomial by each term of the other polynomial. 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.



Multiply (2x – 4)(7x + 5)

(2x – 4)(7x + 5) =

= 14x2 + 10x – 28x – 20

2x(7x) + 2x(5) – 4(7x) – 4(5)

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

Example:


Square of a Binomial

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

(a – b)2 = a2 – 2ab + b2

Product of 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


We can use function notation to represent polynomials.

For example, P(x) = 2x3 – 3x + 4.

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

Find the value P(2) = 2x3 – 3x + 4.

= 2(2)3 – 3(2) + 4P(2)

= 2(8) + 6 + 4

= 6

Evaluating Polynomials

Example:


Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values.

If f(x) = 2x2 + 3x – 4, find f(a + 3).

We replace the variable x with a + 3 in the polynomial function.

f(a + 3) = 2(a + 3)2 + 3(a + 3) – 4

= 2(a2 + 6a + 9) + 3a + 9 – 4

= 2a2 + 12a + 18 + 3a + 9 – 4

= 2a2 + 15a + 23

Evaluating Polynomials

Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 1 § 5.2 More Work with Exponents and Scientific Notation