GED Mathematics · Chapter 5 · pp. 241–317

Algebra Basics, Expressions, and Polynomials

This is where arithmetic turns into algebra. You will work with negative numbers, powers and roots, then learn to read and write expressions with letters in them, and finish by combining and multiplying polynomials. A big share of the GED® Mathematical Reasoning test uses these skills.

Learning objectives

By the end of this chapter you can:

Add, subtract, multiply and divide positive and negative numbers, and find absolute values.
Work out powers and square roots, and write very large and very small numbers in scientific notation.
Follow the order of operations, by hand and on the TI-30XS MultiView™.
Turn words into algebraic expressions, simplify them, and find their value.
Name the parts of a polynomial, and add, subtract, multiply and divide polynomials.

In this chapter: Lesson 1 The number line and signed numbers · Lesson 2 Powers and roots · Lesson 3 Scientific notation · Lesson 4 Order of operations · Lesson 5 Absolute value · Lesson 6 Algebraic expressions · Lesson 7 Expressions and calculator skills · Lesson 8 Understand polynomials · Lesson 9 Simplify polynomials · Lesson 10 Add and subtract polynomials · Lesson 11 Multiply polynomials · Lesson 12 Divide polynomials

🎯 About the test

The test expects you to handle signs, exponents and the order of operations quickly, often without a calculator. Polynomial questions are usually multiple choice, so a good habit is to check a choice by putting a small number, such as x = 2, into both the question and the answer.

✏️ How to use this page

The tabs group lessons that belong together, so Lessons 1 and 5 share a tab. Each lesson has worked examples and a short Try it. There are two tools to practise with: a scientific-notation converter and an order-of-operations stepper. The book's practice sets are in the printable assignment, and there is a quiz at the end.

Next: Lessons 1 and 5, signed numbers →

Lesson 1 · The number line and signed numbers

🔑 Key ideas

Same signs: add the numbers and keep the sign.
Different signs: find the difference, and use the sign of the number further from zero.
To subtract, add the opposite: change − to +, and change the sign of the number after it.

Signed numbers are the positive numbers, the negative numbers and zero. Zero is neither positive nor negative. A number with no sign is positive; a negative number always has a minus sign in front. The sign tells you which direction from zero, and the number tells you how far: −5 is five steps left of zero, and +4 is four steps right.

1 + (−4) = −3: start at 1 and move 4 steps in the negative direction (left).

Adding and subtracting

📘 Example 1 · 15 + (−25)

The signs are different, so subtract: 25 − 15 = 10. The number further from zero is −25, so the answer is negative: 15 + (−25) = −10.

📘 Example 2 · 3 − 8

Add the opposite: 3 − 8 becomes 3 + (−8). Different signs, so 8 − 3 = 5 and the sign is negative: 3 − 8 = −5.

📘 Example 3 · (−5) + 6 − 4 − (−2)

Turn every subtraction into adding the opposite: (−5) + 6 + (−4) + 2.
Add the positives: 6 + 2 = 8. Add the negatives: (−5) + (−4) = −9.
Combine: 8 + (−9) = −1.

💡 On the TI-30XS, the (−) key at the bottom makes a number negative. Use the − key beside + to subtract.

Multiplying and dividing

In algebra the × sign is easily mixed up with the letter x, so multiplication is usually shown with a dot or brackets: −5 · 6, −5(6) and (−5)(6) all mean “−5 times 6”. Division is usually written as a fraction: means 20 ÷ (−5).

Signs	Answer	Examples
the same	positive	(−2)(−9) = 18 · (−18) ÷ (−9) = 2
different	negative	4(−25) = −100 · 12 ÷ (−4) = −3

📘 Example 4 · (−5)(6)(−1)(−2)(2)

Multiply the numbers without their signs: 5 × 6 × 1 × 2 × 2 = 120.
Count the negative factors: there are three. Each pair of negatives makes a positive, and one is left over, so the answer is negative: −120.

💡 An even number of negative factors gives a positive product. An odd number gives a negative one.

✏️ Try it

11 − (−2)
−15 + (−7)
(−18) ÷ (−9)
(−3)(−5)(2)(−10)
At noon it was 92°F. Over two hours the temperature changed by +12°, −5°, +6°, −3° and +13°. What was it at the end?

Answers

1. 11 + 2 = 13 · 2. −22 · 3. 2 · 4. −300 (three negatives) · 5. 92 + 12 − 5 + 6 − 3 + 13 = 115°F

Lesson 5 · Absolute value

🔑 Key ideas

The absolute value of a number is its distance from zero, so it is never negative.
It is written with two straight bars: |−5| = 5 and |5| = 5.
In the order of operations, treat the bars like brackets: work out what is inside first.

📘 Example 5 · −20|17 − 34|

Inside the bars first: 17 − 34 = −17.
Take the absolute value: |−17| = 17.
Multiply: −20 × 17 = −340. The bars only make the inside positive; the −20 in front keeps its sign.

You use absolute value in everyday life without calling it that. If you have $314 in the bank and write a check for $400, the bank shows your balance as −$86, but you think “I'm $86 short”. That $86 is the absolute value.

✏️ Try it

5 + |−6|
−|110 − 201|
|−6| − |−7|
At 9 p.m. it was −5°F. By 5 a.m. it was −15°F. By how many degrees did the temperature change?

Answers

1. 5 + 6 = 11 · 2. −|−91| = −91 · 3. 6 − 7 = −1 · 4. |−15 − (−5)| = |−10| = 10 degrees (it fell by 10)

Next: Lessons 2 and 3, powers, roots and scientific notation →

Lesson 2 · Powers and roots

🔑 Key ideas

In 2³, 2 is the base and 3 is the exponent.
To find a power, write the base as many times as the exponent says, then multiply.
To find a square root, ask “what number times itself gives this number?”

A power is a short way to write repeated multiplication. 5 × 5 × 5 × 5 is “five to the fourth power”, written 5⁴. The small raised number, the exponent, says how many times the base appears.

📘 Example 1 · Find 2⁵

2⁵ = 2 × 2 × 2 × 2 × 2 = 32. (Not 2 × 5 = 10, which is the most common mistake.)

Special exponents to memorise

Rule	Example
A number to the power 1 is itself.	8¹ = 8
Any number except 0 to the power 0 is 1.	6⁰ = 1
A negative exponent makes a fraction with 1 on top.	2⁻³ = =
A negative base needs brackets to be squared.	(−3)² = 9, but −3² = −9

💡 −3² means “the negative of 3²”, because the exponent is done before the minus sign. Only (−3)² squares the −3.

Powers on the calculator

Task	Keys	Display shows
The square of 24	`clear` 24 `x²` `enter`	576
9⁴	`clear` 9 `^` 4 `enter`	6561
(−3)⁴	`clear` `(` `(−)` 3 `)` `^` 4 `enter`	81
√90	`clear` `2nd` `x²` 90 `enter`	3√10; press the toggle key `◄ ►` to see 9.486832981

Without the brackets, (−) 3 ^ 4 gives −81, because the calculator follows the order of operations.

Square roots

Squaring is multiplying a number by itself: 6² = 36. Going back the other way is the square root: √36 = 6. (−6)(−6) is also 36, but the GED® test only asks for the positive root.

Squares from 1 to 15. Learn these by heart; they come up again in geometry.
n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n²	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

📘 Example 2 · Estimate √90

9 × 9 = 81 and 10 × 10 = 100. 90 is between them, so √90 is between 9 and 10. The calculator gives 9.486…, which is 9.5 to the nearest tenth.

✏️ Try it

5³
25⁰
4⁻²
(−2)⁸
Which has the least value: 3⁻³, 4⁰, 4¹ or 2⁻⁴?

Answers

1. 125 · 2. 1 · 3. · 4. 256 (eight negatives make a positive) · 5. 3⁻³ = , which is smaller than 2⁻⁴ =

Lesson 3 · Scientific notation

🔑 Key ideas

In scientific notation, a number from 1 up to (but not including) 10 is multiplied by a power of ten.
Very small numbers use a negative power of ten.
When comparing, the number with the greater power of ten is greater.

Power	10⁴	10³	10²	10¹	10⁰	10⁻¹	10⁻²	10⁻³	10⁻⁴
Value	10,000	1,000	100	10	1	0.1	0.01	0.001	0.0001

Spot the pattern: a positive exponent is the number of zeros after the 1, and a negative exponent is the number of decimal places. That means the exponent tells you how many places to move the decimal point.

Change	Do this	Example
scientific → standard, positive power	Move the point right.	3.7 × 10⁶ = 3,700,000
scientific → standard, negative power	Move the point left.	5.173 × 10⁻⁴ = 0.0005173
standard → scientific, big number	Move the point left until one digit is in front. Count the places: that is the power.	2300 = 2.3 × 10³
standard → scientific, small number	Move the point right until one non-zero digit is in front. The power is negative.	0.00042 = 4.2 × 10⁻⁴

📘 Example 3 · Which is greater: 4.5 × 10³ or 9.8 × 10⁴?

Just compare the powers of ten. 10⁴ = 10,000 is bigger than 10³ = 1,000, so 9.8 × 10⁴ is greater (98,000 against 4,500).

💡 If the answer choices are in scientific notation, change your answer into scientific notation rather than changing all four choices into standard form.

🧪 Try it: scientific-notation converter

Type a standard number such as 23,500 or 0.0095, or scientific notation such as 5.173e-4 or 3.7 x 10^6.

✏️ Try it

Write 12,400,000 in scientific notation.
Write 0.0095 in scientific notation.
Write 1.7 × 10⁻⁵ in standard form.
Many satellites orbit about 23,500 miles above Earth. Write that in scientific notation.

Answers

1. 1.24 × 10⁷ · 2. 9.5 × 10⁻³ · 3. 0.000017 · 4. 2.35 × 10⁴ miles

Next: Lesson 4, order of operations →

Lesson 4 · Order of operations

🔑 Key ideas

Everyone does the operations in the same order, so everyone gets the same answer.
With brackets inside brackets, start with the innermost set and work outwards.
A fraction bar is a grouping symbol: work out the top and the bottom first, then divide.

Parentheses and other grouping symbols: ( ), [ ], { }, | |, and the fraction bar
Exponents and roots
Multiplication and Division, from left to right
Addition and Subtraction, from left to right

💡 “PEMDAS” helps you remember the order, but × and ÷ are one step, done left to right. So are + and −. Multiplication does not always come before division.

📘 Example 1 · + 3(−1)

Brackets: 5 + 3 = 8. Now you have + 3(−1).
Exponent: 8² = 64, giving + 3(−1).
Divide and multiply, left to right: 64 ÷ 4 = 16, and 3(−1) = −3.
Add: 16 + (−3) = 13.

The (−1) has brackets around it, but there is nothing to work out inside them. They are only there to show that 3 is multiplied by −1.

📘 Example 2 · 4[5(−4 + 3) + 2]

Innermost brackets: −4 + 3 = −1, giving 4[5(−1) + 2].
Inside the square brackets, multiply before adding: 5(−1) = −5, then −5 + 2 = −3, giving 4[−3].
Multiply: 4 × (−3) = −12.

📘 Example 3 · + 6

Top: 12 + 8 = 20. Bottom: 9 − 4 = 5. Divide: 20 ÷ 5 = 4. Then add: 4 + 6 = 10.

🧪 Try it: order-of-operations stepper

Use numbers, + - * / (or × ÷), ^ for a power, brackets ( ) [ ] and | | for absolute value. Write a number next to a bracket to multiply, as in 4(3). For a fraction bar, put brackets round the top and the bottom: (12 + 8)/(9 - 4).

Try:

✏️ Try it

16 ÷ (10 − 6)²
5² − (5 − 7)(2)
2³ + (8 − 5)² − 3
(4 + 2)² + (7 − 2)³
22 + 6[(14 − 5) ÷ 3(17 − 14)]

Answers

1. 16 ÷ 16 = 1 · 2. 25 − (−4) = 29 · 3. 8 + 9 − 3 = 14 · 4. 36 + 125 = 161 · 5. Inside: 9 ÷ 3 × 3 = 9 (left to right), so 22 + 54 = 76

Next: Lessons 6 and 7, algebraic expressions →

Lesson 6 · Algebraic expressions

🔑 Key ideas

An algebraic expression uses numbers, operations and variables to show how numbers are related.
A variable is a letter that stands for an unknown number. In one expression, the same letter always means the same number.
You can only add or subtract like terms.

Writing expressions from words

In words	In symbols
the product of 5 and a number	5x
a number decreased by 12	x − 12
the sum of 3 and the square of a number	3 + x²
6 less than the quotient of a number and 2	− 6
one-half a number increased by 15	x + 15
4 times the difference of −3 and a number	4(−3 − x)
a number less another number	x − y
10 less the square root of the sum of a number and 3	10 − √(x + 3)

💡 “Less than” flips the order: “6 less than x” is x − 6, not 6 − x.

📘 Example 1 · A setup problem

Shipping on an online book order costs 4% of the total cost c plus $0.95 for each of the n books. Which expression gives the shipping? The 4% of c is 0.04c, and $0.95 for each book is 0.95n. So the shipping is 0.04c + 0.95n.

Check with easy numbers: a $100 order of 2 books gives 0.04(100) + 0.95(2) = 4 + 1.90 = $5.90, which is sensible.

Simplifying expressions

A term is a number, a variable, or numbers and variables multiplied or divided together, such as 7, x, 3x² or . Like terms have the same variable with the same exponent: 3x² and 5x² are like terms, but 6x and 2x² are not. To simplify, combine the like terms.

📘 Example 2 · Simplify 2x − 5 + 4x² − 8 + 6x

The x terms: 2x + 6x = 8x. The numbers: −5 − 8 = −13. The x² term stays. Write the highest power first: 4x² + 8x − 13.

The distributive property removes brackets: a(b + c) = ab + ac. Multiply every term inside by the term outside, and carry its sign with it.

📘 Example 3 · Simplify 4x − 3(x + 9) + 15

Think of the − as part of the −3: 4x + (−3)(x + 9) + 15.
Distribute: (−3)(x) = −3x and (−3)(9) = −27, giving 4x − 3x − 27 + 15.
Combine like terms: x − 12. (1x is written just x.)

💡 A minus sign in front of brackets is like a −1: −(x − 3) = −x + 3.

Evaluating expressions

Evaluate means find the value. Put the given number in place of each variable, then follow the order of operations.

📘 Example 4 · Find when x = 6 and y = 5

Substitute: = = 5.

Factors and multiples

A prime number has exactly two positive factors, 1 and itself: 2, 3, 5, 7, 11, 13… (1 is not prime). The prime factorization of a number is the primes that multiply to make it: 12 = 2 × 2 × 3.

Find	How	Example
Greatest common factor (GCF)	Multiply the prime factors the numbers share.	40 = 2·2·2·5, 140 = 2·2·5·7. Shared: 2·2·5, so GCF = 20.
Least common multiple (LCM)	Take each prime the most times it appears in either number.	20 = 2·2·5, 30 = 2·3·5. Take 2·2·3·5, so LCM = 60.

The GCF lets you factor an expression: 40 + 140 = 20(2 + 7). The LCM gives the lowest common denominator when you add fractions.

✏️ Try it

Write “the product of 8 and a number is decreased by 10” in symbols.
Simplify 4(y + 8) + 3(y − 6).
Simplify 5 − (x − 3) + 4x.
Find the value of y² + 16 − (y − 5)² when y = 3.
Find the GCF and LCM of 45 and 75.

Answers

1. 8x − 10 · 2. 4y + 32 + 3y − 18 = 7y + 14 · 3. 5 − x + 3 + 4x = 3x + 8 · 4. 9 + 16 − (−2)² = 21 · 5. GCF 15, LCM 225

Lesson 7 · Expressions and calculator skills

🔑 Key ideas

The TI-30XS follows the order of operations by itself.
Use the (−) change-sign key to enter a negative number.
Use the bracket keys ( and ) when you need a different order.

A calculator that follows the order of operations uses algebraic logic. Test yours: press 2 + 3 × 4 enter. If it shows 14, it multiplied first and is fine for the test. If it shows 20, it worked left to right, so practise on a different one.

📘 Example 5 · 2x² + 3x − 5 when x = −4

Press: clear 2 × ( (−) 4 ) x² + 3 × (−) 4 − 5 enter

The display shows 15. By hand: 2(16) + 3(−4) − 5 = 32 − 12 − 5 = 15.

The brackets round −4 matter. Without them, (−) 4 x² gives −16, not 16.

📘 Example 6 · 3(x + 6) when x = 4

Press: clear 3 ( 4 + 6 ) enter. The calculator waits for the closing bracket before it adds, and the display shows 30.

📘 Example 7 · 2x + 1 + √(50x + 25) when x = 4

You can do part of it by hand: 2(4) + 1 = 9. Under the root: 50(4) + 25 = 225, and 2nd x² 225 enter gives 15. Add: 9 + 15 = 24.

💡 Check calculator work by entering it a second time. If you get the same answer twice, you probably pressed the keys you meant to.

✏️ Try it

What is 5x² + 3x + 5 when x = 2?
If x = −3, what is 7x² + 2x − 6?
What is 2(x² + 6) + 3(x − 1) when x = 5?
If x = −5 and y = 2, which is greatest: x + y, −x + y, xy or −2xy?

Answers

1. 20 + 6 + 5 = 31 · 2. 63 − 6 − 6 = 51 · 3. 62 + 12 = 74 · 4. −2xy = 20 (the others are −3, 7 and −10)

Next: Lessons 8 to 10, polynomials →

Lesson 8 · Understand polynomials

🔑 Key ideas

A polynomial is one term or a sum of terms made from coefficients, variables, whole-number exponents and constants.
A monomial has one term, a binomial two, and a trinomial three.
A variable with no number in front has a coefficient of 1.

Take the polynomial 4x³ + 3x² − 5:

Part	Meaning	In 4x³ + 3x² − 5
term	a piece separated by + or −	4x³, 3x² and −5
coefficient	the number in front of the variables	4 and 3
variable	the letter	x
exponent	the power on a variable	3 and 2
constant	a number with no variable; it never changes	−5 (the sign belongs to it)
degree	the largest exponent (with one variable)	3

Type	Terms	Examples
monomial	1	4xy² · 25a · y²
binomial	2	3x + 2 · g + h
trinomial	3	4x + 3y² − 4 · x² + 14x + 3

Some expressions are not polynomials: dividing by a variable (), a negative exponent (4x⁻²), a fractional exponent (x^1/2) or a variable under a root (√x). Dividing by a number is fine (), and so is a root of a number (√2 x).

✏️ Try it

Is 2x⁴ + 3x + 4 a monomial, binomial or trinomial?
Is 2xy²z a monomial, binomial or trinomial?
What is the sum of the coefficients in a − 3b − c + 2d?

Answers

1. Trinomial (three terms) · 2. Monomial (it is all one product) · 3. 1 − 3 − 1 + 2 = −1

Lesson 9 · Simplify polynomials

🔑 Key ideas

Like terms can be combined; unlike terms cannot.
Like terms have exactly the same variables with the same exponents. Numbers on their own are like terms too.
To combine like terms, add their coefficients and keep the variable part.

8x³ and 4x³ are like terms. 3a and 5b are not, and neither are 9x and 7x². So 7x² + 4x² = 11x², but 7x² + 4x³ cannot be simplified.

📘 Example 1 · 4y³ + 2x³ − 7y³

Only the two y³ terms are alike: 4y³ − 7y³ = −3y³. The answer is −3y³ + 2x³.

📘 Example 2 · 7x⁴ − 13x² + x⁴ + 2x² + 5x + 3 + x

Work down from the highest power.

x⁴: 7x⁴ + x⁴ = 8x⁴
x²: −13x² + 2x² = −11x²
x: 5x + x = 6x. The constant 3 stays.

The answer is 8x⁴ − 11x² + 6x + 3.

💡 Treat the sign in front of a term as part of it. Then the signed-number rules from Lesson 1 do the rest.

✏️ Try it

Are 2x²y and 8x²y like terms? Are x²y and xy²?
14ab + ab² + 2ab + 3
−3x² + 6x − 2x² − 10x + 5
8x² − 4x + 7 + 4x³ − x² + 7x − 2

Answers

1. Yes; no (the exponents sit on different letters) · 2. ab² + 16ab + 3 · 3. −5x² − 4x + 5 · 4. 4x³ + 7x² + 3x + 5

Lesson 10 · Add and subtract polynomials

🔑 Key ideas

To add polynomials, drop the brackets and combine like terms.
To subtract, change the sign of every term in the second polynomial, then add.
Simplify inside each bracket first if you can.

📘 Example 1 · (3x² + x + 4) + (2x² + 2x − 16)

3x² + 2x² = 5x² · x + 2x = 3x · 4 + (−16) = −12. The sum is 5x² + 3x − 12.

📘 Example 2 · (3x² + x + 4) − (7x² + 2x − 16)

Change every sign in the second bracket: −7x² − 2x + 16.
Drop the brackets: 3x² + x + 4 − 7x² − 2x + 16.
Combine: −4x² − x + 20.

📘 Example 3 · (x² + 2x + 3) − (4x² − x + 6)

Changing the signs turns −x into +x: x² + 2x + 3 − 4x² + x − 6 = −3x² + 3x − 3.

💡 The usual mistake is changing only the first sign in the second bracket. Change all of them.

More than one variable

The method is the same; just be strict about like terms. 2xy and 7xy combine to 9xy, but 2xy and 7x cannot be combined.

📘 Example 4 · (3x² + 6x + 3xy) + (5x² − 6xy + 14y)

3x² + 5x² = 8x² · 3xy − 6xy = −3xy · 6x and 14y have no partners. The sum is 8x² + 6x − 3xy + 14y.

📘 Example 5 · (3xy² + x + 2xy + 4) − (7xy² + 2xy − 16)

3xy² + x + 2xy + 4 − 7xy² − 2xy + 16 = −4xy² + x + 20. The xy terms cancel: 2xy − 2xy = 0.

✏️ Try it

(5x² − 3x − 4) + (3x² − 2x + 6)
(x + 16) − (4x + 3)
(17a² − 4a − 4) − (16a² + 6a − 6)
(x + y + z) − (−2x + 2y + z)

Answers

1. 8x² − 5x + 2 · 2. −3x + 13 · 3. a² − 10a + 2 · 4. 3x − y

Next: Lessons 11 and 12, multiply and divide →

Lesson 11 · Multiply polynomials

🔑 Key ideas

Multiply the coefficients, then add the exponents of the same variable: x² · x³ = x⁵.
To multiply by a monomial, multiply it by every term in the brackets.
To multiply two binomials, use FOIL: First, Outer, Inner, Last.

Monomial × monomial

A letter with no exponent has an exponent of 1, so x(x³) = x¹ · x³ = x⁴.

📘 Example 1 · (2ab)(4b)

Coefficients: 2 × 4 = 8. Variables: a stays, and b¹ · b¹ = b². The product is 8ab².

Monomial × binomial

📘 Example 2 · −2z²(z + 3yz)

−2z² × z = −2z³, and −2z² × 3yz = −6yz³. The product is −2z³ − 6yz³.

Binomial × binomial: FOIL

📘 Example 3 · (2g² + 9)(3g² − 4)

Step	Multiply	Result
First	2g² × 3g²	6g⁴
Outer	2g² × (−4)	−8g²
Inner	9 × 3g²	27g²
Last	9 × (−4)	−36

Combine the like terms: −8g² + 27g² = 19g². The product is 6g⁴ + 19g² − 36.

💡 Always finish FOIL by combining like terms. The Outer and Inner products are often alike.

📘 Example 4 · Two patterns worth knowing

(x + y)(x − y) = x² − xy + xy − y² = x² − y². The middle terms cancel, so x² − 25 = (x + 5)(x − 5).

(x + y)(x + y) = x² + 2xy + y², not x² + y².

✏️ Try it

(5xyz)(2xy²z⁴)
−5ab(3b + 11c)
(x + 5)(x − 6)
(3x + 3)(3x + 5)
(4ab + 2)(3ab − 7)

Answers

1. 10x²y³z⁵ · 2. −15ab² − 55abc · 3. x² − x − 30 · 4. 9x² + 24x + 15 · 5. 12a²b² − 22ab − 14

Lesson 12 · Divide polynomials

🔑 Key ideas

A polynomial written as a fraction is simplified by dividing the top and bottom by a common factor.
The common factor can be a number, a variable, or a whole expression. You may need to factor it out first.
When dividing, subtract the exponents of the same variable: x⁵ ÷ x² = x³.

📘 Example 1 · Divide by a number:

Both terms on top divide by 3. Factor it out: , and the 3s cancel: 2x + 3.

📘 Example 2 · Divide by an expression:

The common factor is 4x: = x + 2.

📘 Example 3 · Split the fraction:

Give each term on top its own copy of the bottom: − = 2x − 3. Splitting makes it clear which parts cancel.

💡 You can only cancel a factor that multiplies the whole top and the whole bottom. In , the 2s do not cancel.

Undefined expressions

You can never divide by zero. If the bottom of a fraction equals 0, the expression is undefined. For example, is undefined when x = y, because then y − x = 0.

✏️ Try it

James has 27x apples, Rachel 51y oranges and Glen 60z peaches. What is the average number of pieces of fruit they have?
For what value of a is undefined?

Answers

1. 2x + 3 · 2. 2a + 3 · 3. 3m³ − 2m + 1 · 4. (27x + 51y + 60z) ÷ 3 = 9x + 17y + 20z · 5. a = 4

Next: summary and quiz →

Chapter summary

Signed numbers: same signs, add and keep the sign; different signs, subtract and take the sign of the number further from zero. To subtract, add the opposite. When multiplying or dividing, same signs give a positive answer and different signs a negative one.
Absolute value is distance from zero, so it is never negative. Work inside the bars first, like brackets.
Powers: the exponent counts how many times the base is multiplied. a¹ = a, a⁰ = 1, a negative exponent makes a fraction, and (−3)² = 9 but −3² = −9.
Scientific notation is a number from 1 to just under 10 times a power of ten. The exponent counts how many places the decimal point moves.
Order of operations: grouping symbols, exponents and roots, × and ÷ from left to right, then + and − from left to right.
Expressions: translate words carefully (“less than” flips the order), distribute to remove brackets, combine like terms, and substitute to evaluate. On the TI-30XS, put negative numbers in brackets before squaring them.
Polynomials: add by combining like terms; subtract by changing every sign in the second one; multiply by adding exponents and using FOIL; divide by cancelling a common factor. A fraction whose bottom is zero is undefined.

📄 Practice questions

The book's own practice sets for Lessons 1–12 and its chapter practice questions are in the printable assignment for this chapter.

Mini-quiz

1. What is −7 + 3?

2. What is (−6)(−4)(−1)?

3. Which number is equal to 2⁻³?

4. Which is 0.00042 written in scientific notation?

5. Using the order of operations, what is 3 + 4 × 2²?

6. What is −20|17 − 34|?

7. Which expression means “6 less than the product of 5 and a number”?

8. On the TI-30XS, which key do you press to enter a negative number such as −4?

9. Simplify: 4x − 3(x + 9) + 15

10. Simplify: (3x² + x + 4) − (7x² + 2x − 16)

11. Multiply using FOIL: (x + 5)(x − 6)

12. Simplify: (10x³ − 15x²) ÷ (5x²)

Source: GED® Test Prep Mathematics, Chapter 5 (Algebra Basics, Expressions, and Polynomials). Explanations, the Try it questions' answers, some of the examples, the tools and the quiz are Guru Jazzy's own. GED® is a registered trademark of the American Council on Education; this page is not made or endorsed by GED Testing Service.