Файловый архив студентов. Файловый архив студентов.
1265 вузов, 4638 предметов.
/ Регистрация
FAQ Обратная связь Вопросы и предложения
Добавил:
Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Московский государственный физико-технический университет (МФТИ)
Предмет:
[НЕСОРТИРОВАННОЕ]
Файл:

GRE_Math_Bible_eBook

.pdf
Скачиваний:
311
Добавлен:
27.03.2016
Размер:
5.6 Mб
Скачать
☆

Factoring 321

5.We have the equation x – 3 = 10/x. Multiplying the equation by x yields x2 – 3x = 10. Subtracting 10 from both sides yields x2 – 3x – 10 = 0. Factoring the equation yields (x – 5)(x + 2) = 0. The possible solutions are 5 and –2. The only solution that also satisfies the given inequality x > 0 is x = 5. The answer is

(D).

6.Adding 1 to both sides of the given equation x2 – 4x + 3 = 0 yields x2 – 4x + 4 = 1. Expanding (x – 2)2 by

the Difference of Squares formula (a – b)2 = a2 – 2ab + b2 yields x2 – 4.x + 22 = x2 – 4x + 4 = 1. Hence, (x – 2)2 = 1, and the answer is (C).

Algebraic Expressions

A mathematical expression that contains a variable is called an algebraic expression. Some examples of algebraic expressions are x2 , 3x – 2y, 2z( y3 z12 ). Two algebraic expressions are called like terms if both the variable parts and the exponents are identical. That is, the only parts of the expressions that can differ

are the coefficients. For example, 5y3 and

3

y 3

are like terms, as are x + y2

and 7(x + y2 ). However,

2

 

 

 

 

x3 and y3 are not like terms, nor are x – y and 2 – y.

ADDING & SUBTRACTING ALGEBRAIC EXPRESSIONS

Only like terms may be added or subtracted. To add or subtract like terms, merely add or subtract their coefficients:

x2 + 3x2 = (1 + 3)x2 = 4x2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 x 5 x = (2 5) x = 3 x

 

 

 

 

 

 

 

 

 

1 2

 

 

 

1

2

 

1 2

 

1 2

.5

x +

 

+.2

x +

 

 

= (.5

+.2) x +

 

 

=. 7 x +

 

 

 

y

 

 

 

 

 

y

 

 

 

y

 

 

 

 

 

y

(3x3 + 7x 2 + 2x + 4)+ (2x 2 2x 6)= 3x3 + (7+ 2)x2 + (2 2)x + (4 6) = 3x3 + 9x 2 2

You may add or multiply algebraic expressions in any order. This is called the commutative property:

x + y = y + x

xy = yx

For example, –2x + 5x = 5x + (–2x) = (5 – 2)x = 3x and (x – y)(–3) = (–3)(x – y) = (–3)x – (–3)y = –3x + 3y.

Caution: the commutative property does not apply to division or subtraction: 2 = 6 3 3 6 = 12 and 1 = 2 3 3 2 = 1.

When adding or multiplying algebraic expressions, you may regroup the terms. This is called the associative property:

x + (y + z) = (x + y) + z

x(yz) = (xy)z

Notice in these formulas that the variables have not been moved, only the way they are grouped has changed: on the left side of the formulas the last two variables are grouped together, and on the right side of the formulas the first two variables are grouped together.

322

Algebraic Expressions 323

For example, (x – 2x) + 5x = (x + [–2x]) + 5x = x + (–2x + 5x) = x + 3x = 4x and

2(12x ) = (2 12)x = 24x

The associative property doesn't apply to division or subtraction: 4 = 8 ÷ 2 = 8 ÷ (4 ÷ 2) (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1 and

–6 = –3 – 3 = (–1 – 2) – 3 –1 – (2 – 3) = –1 – (–1) = –1 + 1 = 0.

Notice in the first example that we changed the subtraction into negative addition: (x – 2x) = (x + [– 2x]). This allowed us to apply the associative property over addition.

PARENTHESES

When simplifying expressions with nested parentheses, work from the inner most parentheses out: 5x + (y – (2x – 3x)) = 5x + (y – (–x)) = 5x + (y + x) = 6x + y

Sometimes when an expression involves several pairs of parentheses, one or more pairs are written as brackets. This makes the expression easier to read:

2x(x – [y + 2(x – y)]) = 2x(x – [y + 2x – 2y]) = 2x(x – [2x – y]) = 2x(x – 2x + y) = 2x(–x + y) =

2x2 + 2xy

ORDER OF OPERATIONS: (PEMDAS)

When simplifying algebraic expressions, perform operations within parentheses first and then exponents and then multiplication and then division and then addition and lastly subtraction. This can be remembered by the mnemonic:

PEMDAS

Please Excuse My Dear Aunt Sally

This mnemonic isn’t quite precise enough. Multiplication and division are actually tied in order of operation, as is the pair addition and subtraction. When multiplication and division, or addition and subtraction, appear at the same level in an expression, perform the operations from left to right. For example, 6 2 4 = (6 2) 4 = 3 4 = 12. To emphasize this left-to-right order, we can use parentheses

in the mnemonic: PE(MD)(AS).

Example 1:

2 5 33[4 2 + 1]

=

 

 

 

 

 

(

)

 

 

 

 

 

(A) –21

(B) 32

(C) 45

(D) 60

(E) 78

2 5 33[4 2 + 1] =

 

 

 

 

(

(

 

)

 

 

 

 

 

 

)

 

 

 

 

2

5 33[2 +1] =

By performing the division within the innermost parentheses

 

 

(

)

 

 

 

 

 

2

5

33[3] =

By performing the addition within the innermost parentheses

 

2 – (5 – 27[3]) =

By performing the exponentiation

 

 

 

2 – (5 – 81) =

By performing the multiplication within the parentheses

 

 

2 – (–76) =

By performing the subtraction within the parentheses

 

 

 

2 + 76 =

By multiplying the two negatives

 

 

 

 

78

 

 

 

 

The answer is (E).

324GRE Math Bible

FOIL MULTIPLICATION

You may recall from algebra that when multiplying two expressions you use the FOIL method: First, Outer, Inner, Last:

O

F

(x + y)(x + y) = xx + xy + xy + yy

I

L

Simplifying the right side yields (x + y)(x + y) = x 2 + 2xy + y 2 . For the product ( x – y)(x – y), we get (x y)(x y) = x 2 2xy + y 2 . These types of products occur often, so it is worthwhile to memorize the

formulas. Nevertheless, you should still learn the FOIL method of multiplying because the formulas do not apply in all cases.

Examples (FOIL):

(2 y)(x y2 ) = 2x 2y2 xy + yy2 = 2x 2y2 xy + y3

1

 

 

1 1

 

1 1

 

 

 

 

1

= 1

1

xy + 1= 2

1

 

 

 

y

x

 

=

 

x

 

 

 

 

 

xy + y

 

 

 

 

 

xy

 

y

x

x y

 

y

xy

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

xy

1

 

2

1

 

1

 

1

2

 

1

 

 

 

 

2

 

1

 

2

 

 

 

y

=

 

y

 

y

=

 

 

2

y + y

 

=

 

y + y

 

 

 

 

2

2

 

 

4

 

 

 

2

 

 

 

 

 

2

 

 

2

 

 

 

 

 

 

 

 

 

 

DIVISION OF ALGEBRAIC EXPRESSIONS

When dividing algebraic expressions, the following formula is useful:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x + y

x

 

 

 

y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

 

 

+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

z

 

 

z

 

z

 

 

 

This formula generalizes to any number of terms.

 

 

 

 

 

 

 

 

 

 

 

 

Examples:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x2 + y

=

x2

 

+

y

= x

2 1 +

 

y

= x +

y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

x

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x2 + 2y x3

 

 

x2

 

 

2y x 3

= x 2 2

 

2y

x

3 2

= x0

 

 

2y

x = 1+

2y

x

 

 

 

 

 

=

 

 

 

+

 

 

 

 

 

 

+

 

 

+

 

 

 

 

 

 

x2

 

 

 

x2

 

x2

 

x 2

x

2

x

2

 

x2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When there is more than a single variable in the denomination, we usually factor the expression and then cancel, instead of using the above formula.

Example 2:

 

 

x2

2x + 1

 

=

 

 

 

 

x 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(A) x + 1

(B) –x – 1 (C) –x + 1 (D) x – 1

(E) x – 2

 

x2 2x + 1 (x 1)(x 1)

= x 1. The answer is (D).

 

 

 

 

=

 

 

 

 

 

 

x 1

 

 

x 1

 

 

 

 

 

 

 

Algebraic Expressions 325

Problem Set T:

 

 

 

 

 

 

 

 

 

 

Easy

 

 

 

 

 

 

 

 

 

 

 

 

1. If x 3 and x 6, then

2x 2

72

–

2x 2 18

 

=

 

x 6

 

x 3

 

 

 

 

 

 

 

 

 

 

 

 

(A)

3

 

 

 

 

 

 

 

 

 

 

 

(B)

6

 

 

 

 

 

 

 

 

 

 

 

(C)

9

 

 

 

 

 

 

 

 

 

 

 

(D)

12

 

 

 

 

 

 

 

 

 

 

 

(E)

15

 

 

 

 

 

 

 

 

 

 

 

Medium

 

 

 

 

 

 

 

 

 

 

 

 

2.

 

 

 

 

 

Column A

 

 

 

 

 

 

 

 

 

Column B

 

 

 

 

 

(2x 11)(2x + 11)

 

 

 

 

 

 

 

 

(x – 11)(x + 11)

 

 

 

 

 

4

 

 

 

 

 

 

 

 

 

 

3. If

 

3x

 

2, what is the value of

9x 2

4

 

9x

2 4

?

 

 

 

 

 

3x + 2

3x 2

 

 

 

 

 

 

 

 

 

 

 

 

(A)–9

(B)–4

(C)0

(D)4

(E)9

4.

If 1/x + 1/y = 1/3, then

xy

=

 

 

 

 

 

 

 

 

 

x + y

 

(A)

1/5

 

 

 

 

 

 

 

(B)

1/3

 

 

 

 

 

 

 

(C)

1

 

 

 

 

 

 

 

 

(D)

3

 

 

 

 

 

 

 

 

(E)

5

 

 

 

 

 

 

 

5.

If x = 2 and y = –1, which one of the following expressions is greatest?

 

(A)

x + y

 

(B)

xy

 

(C)

–x + y

 

(D)

x – y – 1

 

(E)

–x – y

6.

If b is one-fourth of a, then what is the value of

a

+ b

 

?

 

 

 

 

 

 

 

 

 

 

ab

 

(A)

1/5

 

 

 

 

 

 

 

(B)

1/3

 

 

 

 

 

 

 

(C)

1/2

 

 

 

 

 

 

 

(D)

1

1

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

(E)

2

1

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

326 GRE Math Bible

7.If 3 2x = 1, then what is the value of (3 – 2x) + (3 – 2x)2 ?

(A)0

(B)1

(C)2

(D)3

(E)4

8.

Column A

 

 

 

 

 

 

Column B

 

(1111.0)2 – (999.0)2

 

 

 

 

 

 

(1111.5)2 – (999.5)2

9.

Column A

|x| 1/2

 

 

 

Column B

 

 

4 x 2 1

 

 

 

 

 

 

 

 

4 x 2 1

 

 

 

2x +1

 

 

 

 

 

 

 

2x 1

10.

Column A

x =

 

1

 

 

 

Column B

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

1+

 

 

 

 

 

 

 

 

 

 

1+

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

1

 

Algebraic Expressions 327

Answers and Solutions to Problem Set T

Easy

1. Start by factoring 2 from the numerators of each fraction:

2(x 2 36)

–

2(x 2 9)

x 6

x 3

 

Next, apply the Difference of Squares Formula a2 b2 = (a + b)(a b) to both fractions in the expression:

 

2(x + 6)(x 6)

–

2(x + 3)(x 3)

=

 

 

x 6

x 3

 

 

 

 

Next, cancel the term x – 6 from the first fraction and x – 3 from the second fraction:

2(x + 6) – 2(x + 3) = 2x + 12 – 2x – 6 = 6

Hence, the answer is (B).

 

Medium

 

 

 

2. Applying the Difference of Squares formula (a + b)(a – b) = a2 – b2 yields

 

Column A

 

 

Column B

 

(2x)2 112

 

 

x2 – 112

4

 

 

 

 

 

Column A

 

 

Column B

 

 

4 x 2 121

 

 

 

x2 – 121

4

 

 

 

 

 

Column A

 

 

Column B

 

 

x 2 121

 

 

x2 – 121

4

 

 

 

 

Subtracting x2 from both columns yields

 

 

Column A

 

 

Column B

 

 

–121/4

 

 

–121

Since –121/4 > –121, Column A is greater than Column B and the answer is (A).

3.

9x 2

4

 

9x 2 4

 

 

 

 

 

 

3x + 2

3x 2

 

 

 

 

 

1

1

 

 

 

 

 

 

 

 

 

 

 

 

= (9x 2 4)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3x + 2

3x 2

 

 

= (9x 2 4)

 

 

(3x 2) (3x + 2)

 

 

 

 

 

 

(3x + 2)(3x 2)

 

 

 

 

 

 

 

 

 

 

 

= (9x 2 4)

 

 

3x 2 3x 2

 

 

 

 

 

 

 

 

(3x)2 22

 

 

 

 

 

 

 

 

 

 

 

 

 

= (9x 2 4)

 

4

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

9x

4

 

 

 

 

= –4 The answer is (B).

by factoring out the common term 9x2 – 4

Since |3x| 2, (3x)2 4, and therefore 9x2 – 4 0.

Hence, we can safely cancel 9x2 – 4 from numerator and denominator.

328 GRE Math Bible

4. Multiplying the given equation 1/x + 1/y = 1/3 by xy yields y + x = xy/3, or x + y = xy/3. Multiplying

both sides of the equation x + y = xy/3 by 3/(x + y) yields xy = 3. The answer is (D). x + y

5.Choice (A): x + y = 2 + (–1) = 1. Choice (B): xy = 2(–1) = –2. Choice (C): –x + y = –2 + (–1) = –3.

Choice (D): x – y – 1 = 2 – (–1) – 1 = 2. Choice (E): –x – y = –2 – (–1) = –1.

The greatest result is Choice (D). The answer is (D).

6. We are given that b is 1/4 of a. Hence, we have the equation b = a/4. Multiplying both sides of this equation by 4/b yields 4 = a/b.

Now,

a + b =

 

ab

 

 

 

 

 

 

a

 

 

+

 

b

 

 

=

 

 

 

 

 

 

 

 

 

ab

ab

 

 

 

 

 

 

 

 

 

 

 

 

 

a

2

 

 

+

 

b

2

 

 

=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ab

ab

a2 +b2 = ab ab

a +b = b a

4 +14 = 2 + 12 = 2 12

The answer is (E).

7.We have 3 2x = 1. Squaring both sides of the equation yields (3 – 2 x) = 1. Squaring both sides of the equation again yields (3 – 2x)2 = 1. Hence, (3 – 2x) + (3 – 2x)2 = 1 + 1 = 2. The answer is (C).

8.Column B = (1111.5)2 – (999.5)2 =

= (1111.5 – 999.5)(1111.5 + 999.5)

by the Difference of Squares formula

 

a2 – b2 = (a – b)(a + b)

=(1111 + 0.5 – 999 – 0.5)(1111 + 0.5 + 999 + 0.5)

=(1111 – 999)(1111 + 999 + 1)

=(1111 – 999)(1111 + 999) + (1111 – 999)

= 11112 – 9992 + (1111 – 999)

by the Difference of Squares formula

 

(a – b)(a + b) = a2 – b2

=Column A + (1111 – 999)

=Column A + (a positive number)

From this equation, it is clear that Column B is greater than Column A, by 1111 – 999, a positive number. Hence, the answer is (B).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algebraic Expressions 329

9. Applying the Difference of Squares Formula a2 b2 = (a + b)(a b) to both columns yields

 

 

(2x + 1)(2x 1)

 

 

 

 

 

 

 

 

 

 

 

 

 

(2x + 1)(2x 1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2x + 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2x 1

Reducing yields

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2x – 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2x + 1

Subtracting 2x from both columns yields

 

–1

 

 

 

 

 

 

 

 

 

 

 

1

 

 

The answer is (B).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

x =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1+

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1+

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1+

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

=

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1+

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

3

< 1

 

 

 

 

 

 

 

 

 

 

 

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hence, Column A is less than Column B, and the answer is (B).

Percents

Problems involving percent are common on the GRE. The word percent means “divided by one hundred.” When you see the word “percent,” or the symbol %, remember it means 1001 . For example,

25percent

25 1001 = 14

To convert a decimal into a percent, move the decimal point two places to the right. For example,

0.25 = 25%

0.023 = 2.3%

1.3 = 130%

Conversely, to convert a percent into a decimal, move the decimal point two places to the left. For example,

47% = .47

3.4% = .034

175% = 1.75

To convert a fraction into a percent, first change it into a decimal (by dividing the denominator [bottom] into the numerator [top]) and then move the decimal point two places to the right. For example,

78 = 0.875 = 87. 5%

Conversely, to convert a percent into a fraction, first change it into a decimal and then change the decimal into a fraction. For example,

80% =. 80 = 10080 = 45

Following are the most common fractional equivalents of percents:

33

1

% =

 

1

20% =

1

 

3

3

5

 

 

 

 

 

 

 

66

1 % =

 

2

40% =

2

 

3

5

 

 

3

 

 

 

 

 

25% =

1

 

 

60% =

3

 

4

 

 

5

 

 

 

 

 

 

 

 

50% =

1

 

 

80% =

4

 

2

 

 

5

 

 

 

 

 

 

 

 

330

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]
Помощь Обратная связь Вопросы и предложения Пользовательское соглашение Политика конфиденциальности