Über die Autorin bzw. den Autor

Mary Jane Sterling is a professor of mathematics at Bradley University in Peoria, Illinois. She has been teaching mathematics for over thirty years.

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CliffsNotes Algebra II Practice Pack

John Wiley & Sons

Chapter One

Algebra is a language. You need to know the rules and definitions to understand this language and its many manipulations. In this chapter is a review of some of the important basics of algebra: rules for exponents and operations involving polynomials. These should be reviewed before going on to some of the advanced topics in Algebra II.

Rules for Exponents

A power or exponent tells how many times a number multiplies itself. Many opportunities exist in algebra for combining and simplifying expressions with two or more of these exponential terms in them. The rules used here to combine numbers and variables work for any expression with exponents. They are found in formulas and applications in science, business, and technology, as well as math. The term [a.sup.4] has an exponent of 4 and a base of a. The base is what gets multiplied repeatedly. The exponent tells how many times that repeatedly is.

Laws for Using Exponents

[a.sup.n] x [a.sup.m] = [an.sup.+m] When multiplying two numbers that have the same base, add their exponents.

[a.sup.n]/[a.sup.m] = [a.sup.n-m] When dividing two numbers that have the same base, subtract their exponents.

[([a.sup.n]).sup.m] = [a.sup.n m] When raising a value that has an exponent to another power, multiply the two exponents.

[(a b).sup.n] = [a.sup.n] [b.sup.n] The product of two numbers raised to a power is equal to raising each number to that power and then multiplying them together.

[(a/b).sup.n] = [a.sup.n]/[b.sup.n] The quotient of two numbers raised to a power is equal to raising each of the numbers to that power and then dividing them.

[a.sup.-n] = 1/[a.sup.n] A value raised to a negative power can be written as a fraction with the positive power of that number in the denominator.

[a.sup.0] = 1 Any number (except 0) raised to the 0 power is equal to 1.

Example Problems

These problems show the answers and solutions.

1. Simplify: [x.sup.4] ([x.sup.3/[x.sup.-2]).sup.3]

answer: [x.sup.19]

In this case, the course of action is to simplify the expressions inside the parentheses first, raise that result to the third power, and finally multiply by the first factor.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. Simplify: [y.sup.-3] [([y.sup.2]).sup.4] [y.sup.2]/[yy.sup.3]

answer: [y.sup.5]

The denominator reads [yy.sup.3], which implies that the first factor has an exponent of 1, reading [y.sup.1] [y.sup.3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. Simplify: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

answer: [a.sup.23]/[b.sup.6]

A nice property of fractions is that when they're raised to a negative power, you can rewrite the expression and change the power to a positive if you "flip" the fraction. So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. First, rewrite the second fraction without the negative exponent. Then simplify the fractions inside the parentheses. The next step is to raise the factors in the parentheses to the powers. Lastly, multiply the terms in the two numerators and denominators.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Work Problems

Use these problems to give yourself additional practice.

1. Simplify: [([x.sup.3]/[x.sup.-3]).sup.2]

2. Simplify: [([y.sup.4]/3[x.sup.2]).sup.-3]

3. Simplify: [a.sup.-2] [x.sup.2]/[a.sup.4] x [([a.sup.3]/[x.sup.3]).sup.4]

4. Simplify: [(ab[c.sup.2]).sup.4]/[a.sup.2]b[c.sup.-1]

5. Simplify: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Worked Solutions

1. [x.sup.12] First simplify inside the parentheses. Then raise the result to the second power.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. 27[x.sup.6]/[y.sup.12] First "flip" the fraction and change the power to positive.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. [a.sup.6]/[x.sup.11] First raise the factors in the parentheses to the fourth power. Then simplify the first fraction before multiplying the two fractions together.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. [a.sup.2][b.sup.3][c.sup.9] First raise the numerator to the fourth power. Then simplify the fraction.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. [x.sup.4][z.sup.20]/16 Since both fractions are raised to the fourth power, it is easier to combine them in the same parentheses and then later raise the result to the fourth power.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Adding and Subtracting Polynomials

One major objective of working with algebraic expressions is to write them as simply as possible and in a logical, generally accepted arrangement. When more than one term exists (a term consists of one or more factors multiplied together and separated from other terms by + or -), then you check to see whether they can be combined with other terms that are like them. Numbers by themselves without letters or variables are like terms. You can combine 14 and 8 because you know what they are and know the rules. For instance, 14 + 8 = 22, 14 - 8 = 6, 14(8) = 112, and so on. Numbers can be written so they can combine with one another. They can be added, subtracted, multiplied, and divided, as long as you don't divide by zero. Fractions can be added if they have a common denominator. Algebraic expressions involving variables or letters have to be dealt with carefully. Since the numbers that the letters represent aren't usually known, you can't add or subtract terms with different letters. The expression 2a + 3b has to stay that way. That's as simple as you can write it, but the expression 4c + 3c can be simplified. You don't know what c represents, but you can combine the terms to tell how many of them you have (even though you don't know what they are!): 4c + 3c = 7c. Here are some other examples:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that there are two different kinds of terms, one with the x squared and the other with the y squared. Only those that have the letters exactly alike with the exact same powers can be combined. The only thing affected by adding and subtracting these terms is the coefficient.

Example Problems

These problems show the answers and solutions.

1. Simplify 5x[y.sup.2] + 8x - 9[y.sup.2] - 2x + 3[y.sup.2] - 8x[y.sup.2].

answer: -3x[y.sup.2] + 6x - 6[y.sup.2]

There are three different kinds of terms. First rearrange the terms so that the like terms are together. Then combine the like terms.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. Simplify 3[x.sup.4] - 2[x.sup.3] + [x.sup.2] - x + 5 - 2[x.sup.2] + 3[x.sup.3] + 11.

answer: 3[x.sup.4] + [x.sup.3] - [x.sup.2] - x + 16

Rearrange the terms so that the like ones are together. By convention, you write terms that have different powers of the same variable in either decreasing or increasing order of their powers.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Work Problems

Use these problems to give yourself additional practice.

1. Simplify by combining like terms: [m.sup.2] + 3mn + m + 8 + 9mn - [m.sup.2] + 14m.

2. Simplify by combining like terms: 3a + 4b - 6 + 2a - 11.

3. Simplify by combining like terms: 2[pi][r.sup.2] + 8[pi][r.sup.2] - 6[pi]r + 7.

4. Simplify by combining like terms: a + ab + ac + ad + ae + 1.

5. Simplify by combining like terms: 5[x.sup.2]y - 2[x.sup.2]y + [x.sup.2]y + [xy.sup.2] - 8[y.sup.2].

Worked Solutions

1. 12mn + 15m + 8 Rearrange the terms so that the terms that can be combined are together.

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2. 5a + 4b - 17 3a + 4b - 6 + 2a - 11 = 3a + 2a + 4b - 6 - 11 = 5a + 4b - 17

3. 10[pi][r.sup.2] - 6[pi]r + 7 Only the terms with the [pi][r.sup.2] will combine.

4. a + ab + ac + ad + ae + 1 This is already simplified. None of the terms have exactly the same variables. There's nothing more to do.

5. 4[x.sup.2]y + x[y.sup.2] - 8[y.sup.2] There are three terms with the same variables raised to the same powers. Be very careful with problems like this.

Multiplying Polynomials

Multiplying polynomials requires that each term in one polynomial multiplies each term in the other polynomial. When a monomial (one term) multiplies another polynomial, the distributive property is used, and the result quickly follows. Multiplying polynomials with more than one term can be very complicated or tedious, but some procedures or methods can be used to provide better organization and accuracy. For instance, to multiply a binomial times another binomial, such as (x + 2)(a + 3), or to multiply a binomial times a trinomial, such as (x + 2)(x + y + 3), you can use the distributive property. Distribute the (x + 2) over the other terms.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

None of the terms are alike, so this can't be simplified further.

The FOIL Method

Multiplying two binomials together is a very common operation in algebra. The FOIL method is preferred when multiplying most types of binomials.

The letters in FOIL stand for First, Outer, Inner, and Last. These words describe the positions of the terms in the two binomials. Each term actually will have two different names, because each term is used twice in the process. In the multiplication problem, (x + 2)(a + 3):

The x and the a are the First terms of each binomial. The x and the 3 are the Outer terms in the two binomials. The 2 and the a are the Inner terms in the two binomials. The 2 and the 3 are the Last terms of each binomial.

These pairings tell you what to multiply.

F: multiply x a

O: multiply x 3

I: multiply 2 a

L: multiply 2 3

Add these together, x a + x 3 + 2 a + 2 3 = ax + 3x + 2a + 6. It's the same result, in a slightly different order, as the one obtained previously with distribution.

When using this FOIL method, you'll notice that, when the two binomials are alike-that is they have the same types of terms-the Outer and Inner terms combine, and the result is a trinomial. If they aren't alike, as shown by this last example, then none of the terms in the solution will combine, and you'll have a four-term polynomial.

Example Problems

These problems show the answers and solutions.

1. (x + 3)(x - 8)

answer: [x.sup.2] - 5x - 24

Using FOIL,

F: multiply x x

O: multiply x(-8)

I: multiply 3 x

L: multiply 3(-8)

Add the products together: [x.sup.2] - 8x + 3x - 24 = [x.sup.2] - 5x - 24

2. (3x - 1)(x - y)

answer: 3[x.sup.2] - 3xy - x + y

Using FOIL,

F: multiply 3x x

O: multiply 3x(-y)

I: multiply -1 x

L: multiply -1(-y)

Add the products together: 3[x.sup.2] - 3xy - x + y. Notice that none of the terms combine-they're not quite alike.

3. (3[x.sup.2] - 1)([x.sup.2] - 2)

answer: 3[x.sup.4] - 7[x.sup.2] + 2

Using FOIL,

F: multiply 3[x.sup.2] [x.sup.2]

O: multiply 3[x.sup.2] (-2)

I: multiply -1 [x.sup.2]

L: multiply -1(-2)

Add the products together: 3[x.sup.4] - 6[x.sup.2] - [x.sup.2] + 2 = 3[x.sup.4] - 7[x.sup.2] + 2

Work Problems

Use these problems to give yourself additional practice. Find the products and simplify the answers.

1. (x - 3)(x + 7)

2. (2y + 3)(3y - 2)

3. (6x - 3)(6x - 5)

4. (ax - 1)(bx - 2)

5. ([m.sup.3] - 3)([m.sup.3] + 11)

Worked Solutions

1. [x.sup.2] + 4x 21 Using FOIL, (x - 3)(x + 7) = (x - 3)(x + 7) = [x.sup.2] + 7x - 3x - 21. The middle two terms combine.

2. 6[y.sup.2] + 5y - 6 Using FOIL, (2y + 3)(3y - 2) = 6[y.sup.2] - 4y + 9y - 6. Again, the middle two terms combine.

3. 36[x.sup.2] - 48x + 15 Using FOIL, (6x - 3)(6x - 5) = 36[x.sup.2] - 30x - 18x + 15.

4. ab[x.sup.2] - (2a + b)x + 2 At first, it appears that the middle two terms can't combine. If the a and b had been numbers, you could have added them together. They still can be added and the result multiplies the x: (ax - 1)(bx - 2) = ax bx - 2ax - bx + 2 = ab[x.sup.2] - (2a + b)x + 2.

5. [m.sup.6] + 8[m.sup.3] - 33 Using FOIL, ([m.sup.3] - 3)([m.sup.3] + 11) = [m.sup.6] + 11[m.sup.3] - 3[m.sup.3] - 33.

Special Products

It's nice to have the FOIL method to multiply two binomials together. Unfortunately, no other really handy tricks exist for multiplying other polynomials. Basically, you just distribute the smaller of the two polynomials over the other polynomial. Any polynomials can be multiplied together. The different types of multiplications are classified by the number of terms in the multiplier. Some products, however, are easier to perform because of patterns that exist in them. These patterns largely are due to the special types of polynomials that are being multiplied together. Whenever you can recognize a special situation and can take advantage of a pattern, you'll save time and be less likely to make an error. Here are the special products:

1. (a + b)(a - b) = [a.sup.2] - [b.sup.2] Multiplying the sum and difference of the same two numbers

(Continues...)

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