Introduction to Algebra

Study Sheet

Introduction to Algebra

Variables, expressions, properties, and simplifying

Variables and What They Represent

Concept
The Big Idea

A variable is a letter (like xx, nn, or yy) that stands for a number we do not know yet, or a number that can change. Instead of writing “some number,” we just write a letter. The same letter always stands for the same value within one problem.

Algebra is arithmetic with a placeholder. A variable stands for a number, an expression is a calculation written with it, and evaluating means putting a number in. Translating words into symbols is the skill: "three more than twice a number" is 2n+32n + 3, and the order of the words is not always the order of the symbols.

Why use a letter at all? Because one expression can then describe every case at once. The sentence "the bag holds three more than it did" is true whether the bag started with 55 marbles or 500500; the expression n+3n + 3 captures exactly that sentence and nothing else. The one promise a variable makes is consistency: inside one problem, every nn means the same number.

Example
Reading a Variable
nn + 3 marbles
The unknown amount nn plus three more: whatever nn is, the total is n+3n+3.

If a bag holds nn marbles and you add 33 more, the bag now holds n+3n + 3 marbles. Here nn could be 55 (giving 88) or 1010 (giving 1313)—the letter lets us describe every case at once.

Side note
Where it goes wrong

Reading 4x4x as a two-digit number when x=5x = 5: "4545". Written next to each other, a number and a letter are multiplied, so 4x=4⋅5=204x = 4\cdot 5 = 20.

Tip

Tip: A number written right next to a variable means multiply. So 4x4x means “44 times xx,” never “forty-something.”

Try it
Try it: a variable for a total

A movie ticket costs $9\$9 and a snack costs $s\$s. Write an expression for the cost of one ticket and one snack, then find the cost when s=4s = 4.

Show the answer

The ticket is a fixed $9\$9 and the snack is the unknown $s\$s, so together they cost 9+s9 + s dollars. When s=4s = 4: 9+4=139 + 4 = 13, so $13\$13.

Expressions vs. Equations

Concept
Two Different Things

An expression is a combination of numbers, variables, and operations with no equals sign, such as 3x+73x + 7. An equation says two expressions are equal and does have an equals sign, such as 3x+7=193x + 7 = 19. You simplify or evaluate an expression; you solve an equation.

The difference matters because it decides what you are allowed to do. An expression is a single quantity, so the only honest moves are ones that keep its value the same: rewriting, simplifying, evaluating. An equation is a claim that two quantities are equal, so you may change both sides together, as long as they stay equal, and the goal becomes finding which values make the claim true.

Concept
Vocabulary of an Expression

Consider  5x+2y−8\ 5x + 2y - 8.

  • A term is a piece separated by ++ or −- signs: the terms are 5x5x, 2y2y, and −8-8.
  • A coefficient is the number multiplied by a variable: 55 is the coefficient of xx, and 22 is the coefficient of yy.
  • A constant is a term that is just a number, with no variable: here the constant is −8-8.
Example
Naming the Parts

In 7a−3+a7a - 3 + a: the terms are 7a7a, −3-3, and aa. The coefficient of the first term is 77; the coefficient of aa (the last term) is 11, because aa means 1a1a. The constant is −3-3.

Side note
Where it goes wrong

Calling −8-8 in 5x+2y−85x + 2y - 8 "88". A term carries the sign in front of it, so the constant is −8-8. The same goes for coefficients: in 6−3m6 - 3m, the coefficient of mm is −3-3.

Tip

Tip: A lone variable like aa has a coefficient of 11, and −x-x has a coefficient of −1-1, even though the 11 is invisible.

Try it
Try it: name every part

For 9−4k+k29 - 4k + k^2: list the terms, give the coefficient of kk and of k2k^2, and name the constant. Is it an expression or an equation?

Show the answer

Split at the ++ and −- signs, keeping each sign with its term: the terms are 99, −4k-4k and k2k^2. The coefficient of kk is −4-4 (the sign belongs to it), the coefficient of k2k^2 is 11 (nothing written means 11), and the constant is 99. There is no equals sign, so it is an expression.

Writing Expressions from Word Phrases

Concept
Key Words

Certain words signal each operation:

  • Addition: sum, plus, more than, increased by, total
  • Subtraction: difference, minus, less than, decreased by, fewer
  • Multiplication: product, times, of, twice, double, triple
  • Division: quotient, divided by, per, split equally

Translating is reading the phrase in the order the arithmetic happens, which is not always the order of the words. "77 less than a number" describes starting with the number and then taking 77 away, so the number is written first: x−7x - 7. Subtraction and division are the operations where this matters, because they are the ones where swapping the order changes the answer (x−7≠7−xx - 7 \ne 7 - x).

Example
Phrase to Expression

“The sum of a number and 66” →x+6\rightarrow x + 6. “Three times a number” →3x\rightarrow 3x. “The quotient of a number and 44” →x4\rightarrow \dfrac{x}{4}. “55 more than twice a number” →2x+5\rightarrow 2x + 5.

Side note
Where it goes wrong

Translating "twice the sum of a number and 55" as 2x+52x + 5. "The sum of a number and 55" is one quantity, x+5x + 5, and "twice" applies to all of it: 2(x+5)2(x + 5). Test with x=1x = 1: twice the sum is 2⋅6=122\cdot 6 = 12, while 2x+5=72x + 5 = 7.

Example
The “Less Than” Order Trap

“77 less than a number” means you start with the number and take 77 away: x−7x - 7, not 7−x7 - x. The number you subtract comes after the minus sign. Same idea with “more than”: “44 more than xx” is x+4x + 4.

Tip

Order trap: “less than” and “subtracted from” reverse the order. “33 less than nn” is n−3n - 3. Read carefully!

Try it
Try it: translate carefully

Write expressions for: (a) 1010 fewer than a number nn; (b) the quotient of nn and 33, increased by 11; (c) 33 times the difference of nn and 22.

Show the answer

(a) Start with nn and remove 1010: n−10n - 10. (b) The quotient comes first, then 11 is added: n3+1\dfrac{n}{3} + 1. (c) "The difference of nn and 22" is one quantity, n−2n - 2, and all of it is tripled: 3(n−2)3(n - 2).

Evaluating Expressions by Substitution

Concept
Substitute, Then Compute

To evaluate an expression, replace each variable with its given value and follow the order of operations (PEMDAS). Wrap negative values in parentheses so you do not lose a sign.

Substitution works because a variable is only a stand-in: once its value is known, the expression is plain arithmetic, and the order of operations decides it. The parentheses around a substituted value are not decoration. They keep the value together as one number, so that −3-3 squared is read as (−3)2=9(-3)^2 = 9 and not as −32=−9-3^2 = -9.

Example
Evaluating with a Positive Value

Evaluate 3x+53x + 5 when x=4x = 4.

3x+5=3(4)+5substitute=12+5=17multiply, then add\begin{aligned} 3x + 5 &= 3(4) + 5 &&\text{substitute}\\ &= 12 + 5 = 17 &&\text{multiply, then add} \end{aligned}
Side note
Where it goes wrong

Evaluating x2x^2 at x=−3x = -3 as −32=−9-3^2 = -9. The exponent applies to what it touches, and without parentheses it touches only the 33. Substituting as (−3)2=(−3)(−3)=9(-3)^2 = (-3)(-3) = 9 gives the right value.

Example
Evaluating with an Integer

Evaluate 2a−b2a - b when a=−3a = -3 and b=5b = 5.

2a−b=2(−3)−5substitute with parentheses=−6−5=−11multiply, then subtract\begin{aligned} 2a - b &= 2(-3) - 5 &&\text{substitute with parentheses}\\ &= -6 - 5 = -11 &&\text{multiply, then subtract} \end{aligned}

A third example with an exponent: evaluate x2−2xx^2 - 2x at x=−3x = -3. Substitute with parentheses: (−3)2−2(−3)(-3)^2 - 2(-3). Exponent first: (−3)2=9(-3)^2 = 9. Then the product: 2(−3)=−62(-3) = -6. Finally 9−(−6)=9+6=159 - (-6) = 9 + 6 = 15.

Tip

Tip: Always put a substituted number in parentheses, especially a negative one. Writing 2(−3)2(-3) keeps you from misreading 2−32-3.

Try it
Try it: substitute with care

Evaluate 4p−q24p - q^2 when p=2p = 2 and q=−5q = -5.

Show the answer

Substitute with parentheses: 4(2)−(−5)24(2) - (-5)^2. The exponent comes first: (−5)2=25(-5)^2 = 25. Then the product 4(2)=84(2) = 8. Last, 8−25=−178 - 25 = -17.

The Properties of Operations

Concept
Four Properties to Know
  • Commutative (order can swap): a+b=b+aa + b = b + a and ab=baab = ba.
  • Associative (regroup with parentheses): (a+b)+c=a+(b+c)(a+b)+c = a+(b+c) and (ab)c=a(bc)(ab)c = a(bc).
  • Distributive (multiply across a sum): a(b+c)=ab+aca(b+c) = ab + ac.
  • Identity (leaves a number unchanged): a+0=aa + 0 = a and a×1=aa \times 1 = a.

These properties are why algebra is possible: they are the only moves that are guaranteed never to change a value. Each one can be seen with objects. Three rows of four chairs is the same as four rows of three (commutative); stacking 2+52 + 5 blocks next to 33 is the same pile as 22 next to 5+35 + 3 (associative); and a 66-by-(x+2)(x + 2) rectangle is the 66-by-xx piece plus the 66-by-22 piece (distributive).

Example
Spotting the Property

4+9=9+44 + 9 = 9 + 4 shows the commutative property of addition. (2×5)×3=2×(5×3)(2 \times 5)\times 3 = 2 \times (5 \times 3) shows the associative property of multiplication. 6(x+2)=6x+126(x + 2) = 6x + 12 shows the distributive property. 8×1=88 \times 1 = 8 shows the identity property of multiplication.

Side note
Where it goes wrong

Using a property on subtraction as if it held: (10−4)−3=3(10 - 4) - 3 = 3, but 10−(4−3)=910 - (4 - 3) = 9. Subtraction is not associative, so regrouping it changes the answer. Rewrite subtraction as adding the opposite, 10+(−4)+(−3)10 + (-4) + (-3), and then you may regroup freely.

Tip

Watch out: subtraction and division are not commutative or associative. 8−3≠3−88 - 3 \neq 3 - 8, and 12÷4≠4÷1212 \div 4 \neq 4 \div 12.

Try it
Try it: property or not?

Name the property, or say "not a property" if the statement is false in general: (a) 7⋅(3⋅k)=(7⋅3)⋅k7 \cdot (3 \cdot k) = (7 \cdot 3)\cdot k; (b) x+0=xx + 0 = x; (c) 12÷3=3÷1212 \div 3 = 3 \div 12.

Show the answer

(a) Only the grouping changed, so it is the associative property of multiplication (and it lets you write 21k21k). (b) Adding 00 leaves xx unchanged: the identity property of addition. (c) False: 12÷3=412 \div 3 = 4 but 3÷12=143 \div 12 = \tfrac14. Division is not commutative.

Using the Distributive Property to Expand

Concept
Multiply Every Term Inside
5x15x35total area 5(x + 3) = 5x + 15
One rectangle, two ways to find its area.

To expand, multiply the number outside the parentheses by each term inside, keeping track of signs:

a(b+c)=ab+aca(b−c)=ab−ac.a(b + c) = ab + ac \qquad a(b - c) = ab - ac.

The distributive property is a statement about area. A rectangle 55 tall and x+3x + 3 wide can be measured all at once, 5(x+3)5(x + 3), or as two smaller rectangles, 5x5x and 5⋅35 \cdot 3. The total area is the same either way, so 5(x+3)=5x+155(x + 3) = 5x + 15. That is also why the outside number must reach every term: each term inside the parentheses is one piece of the width, and each piece gets its own rectangle.

Example
Expanding

5(x+3)=5⋅x+5⋅3=5x+155(x + 3) = 5\cdot x + 5\cdot 3 = 5x + 15. 4(2y−7)=4⋅2y−4⋅7=8y−284(2y - 7) = 4\cdot 2y - 4\cdot 7 = 8y - 28. −3(a+6)=−3⋅a+(−3)⋅6=−3a−18-3(a + 6) = -3\cdot a + (-3)\cdot 6 = -3a - 18.   (The negative multiplies both terms.)

Side note
Where it goes wrong

Writing −4(x−3)=−4x−12-4(x - 3) = -4x - 12. The −4-4 multiplies both terms, including −3-3: (−4)(−3)=+12(-4)(-3) = +12. So −4(x−3)=−4x+12-4(x - 3) = -4x + 12. Checking with x=0x = 0 catches it: the left side is −4(−3)=12-4(-3) = 12.

Tip

Tip: A negative in front distributes to every term. Forgetting the second sign is the most common mistake—−2(x−4)=−2x+8-2(x - 4) = -2x + 8, not −2x−8-2x - 8.

Try it
Try it: expand

Expand −2(3m−5)-2(3m - 5) and 12(8y+6)\tfrac12(8y + 6).

Show the answer

Multiply the outside number into each term. −2⋅3m=−6m-2 \cdot 3m = -6m and −2⋅(−5)=10-2 \cdot (-5) = 10, so −2(3m−5)=−6m+10-2(3m - 5) = -6m + 10. Next, 12⋅8y=4y\tfrac12 \cdot 8y = 4y and 12⋅6=3\tfrac12 \cdot 6 = 3, so 12(8y+6)=4y+3\tfrac12(8y + 6) = 4y + 3.

Combining Like Terms

Concept
Only Like Terms Combine

Like terms have the exact same variable part (same letter, same power). You combine them by adding or subtracting their coefficients. Constants are like terms with each other. You may not combine terms with different variables.

Combining like terms is the distributive property read backward: 3x+5x=(3+5)x=8x3x + 5x = (3 + 5)x = 8x. It works only when the variable parts match, because only then is there a common factor to pull out. Three xx's and five xx's are eight xx's, just as three apples and five apples are eight apples; three xx's and five yy's stay as they are.

Example
Combining
xxx+xxxxxxxxxxxxx= 8x
Three xx-bars and five xx-bars are eight xx-bars.

3x+5x=8x3x + 5x = 8x   (like terms: both have xx). 7a−2a+4=5a+47a - 2a + 4 = 5a + 4   (combine the aa-terms; the 44 stays). 6x+3y6x + 3y   cannot be combined—xx and yy are different variables.

Side note
Where it goes wrong

Writing 3x+2=5x3x + 2 = 5x. The 22 is a constant, not a group of xx's, so it cannot join them. With x=1x = 1: 3(1)+2=53(1) + 2 = 5, but 5x=55x = 5 only by coincidence; with x=2x = 2 the two sides are 88 and 1010.

Tip

Key idea: Only like terms combine. Add the coefficients but keep the variable the same: 2n+5n=7n2n + 5n = 7n, never 7n27n^2.

Try it
Try it: combine

Simplify 6a+4−2a+3b−96a + 4 - 2a + 3b - 9.

Show the answer

Group the like terms: the aa-terms 6a−2a=4a6a - 2a = 4a, the bb-term 3b3b (alone), and the constants 4−9=−54 - 9 = -5. So the expression is 4a+3b−54a + 3b - 5.

Simplifying: Distribute, Then Combine

Concept
The Full Strategy

To fully simplify an expression: first use the distributive property to remove any parentheses, then combine all like terms. Work carefully with signs at each step.

The order is forced by the order of operations: the parentheses hide a multiplication, and it has to be done before anything inside can join terms outside. Once the parentheses are gone, every term is exposed and the like terms can be gathered. A quick way to trust the result is to substitute a small number into the original and into the answer; if they differ, a sign went wrong.

Example
Distribute Then Combine

Simplify 2(x+3)+4x2(x + 3) + 4x.

2(x+3)+4x=2x+6+4xdistribute=6x+6combine 2x+4x\begin{aligned} 2(x + 3) + 4x &= 2x + 6 + 4x &&\text{distribute}\\ &= 6x + 6 &&\text{combine } 2x + 4x \end{aligned}
Side note
Where it goes wrong

Simplifying 10−3(x+2)10 - 3(x + 2) as 7(x+2)7(x + 2). Subtracting first ignores the order of operations: the 33 multiplies (x+2)(x + 2) before anything is subtracted. Correct: 10−3x−6=4−3x10 - 3x - 6 = 4 - 3x. Check with x=0x = 0: the original is 10−6=410 - 6 = 4, which matches 4−3x4 - 3x, while 7(0+2)=147(0 + 2) = 14 does not.

Example
A Trickier One

Simplify 5(2a−1)−3(a+4)5(2a - 1) - 3(a + 4).

5(2a−1)−3(a+4)=10a−5−3a−12distribute both=7a−17combine like terms\begin{aligned} 5(2a - 1) - 3(a + 4) &= 10a - 5 - 3a - 12 &&\text{distribute both}\\ &= 7a - 17 &&\text{combine like terms} \end{aligned}

Notice −3-3 multiplied both aa and 44, giving −3a−12-3a - 12.

Tip

Tip: Distribute first, then combine. Line up like terms and watch every sign, especially after a minus sign in front of parentheses.

Try it
Try it: distribute, then combine

Simplify 4(2k−1)−(k−7)4(2k - 1) - (k - 7) and check your answer at k=1k = 1.

Show the answer

The minus sign in front of the second parentheses is a multiplication by −1-1. Distribute: 8k−4−k+78k - 4 - k + 7. Combine: 8k−k=7k8k - k = 7k and −4+7=3-4 + 7 = 3, giving 7k+37k + 3. Check at k=1k = 1: the original is 4(1)−(−6)=104(1) - (-6) = 10 and 7(1)+3=107(1) + 3 = 10.

Going Deeper: Toward Real Algebra

Concept
From Expression to Function

A function is a rule that takes an input and produces exactly one output. Instead of naming the rule “the expression 3x+53x + 5,” we give it a name and write function notation: f(x)=3x+5f(x) = 3x + 5. Read this as “ff of xx.” The letter inside the parentheses is the input; the whole symbol f(x)f(x) stands for the output. Note that f(x)f(x) does not mean ff times xx—it is the single value the rule returns.

Example
Evaluating a Function

Let f(x)=3x+5f(x) = 3x + 5. To find f(4)f(4), substitute 44 for every xx and compute.

f(4)=3(4)+5substitute x=4=12+5=17simplify\begin{aligned} f(4) &= 3(4) + 5 &&\text{substitute } x = 4\\ &= 12 + 5 = 17 &&\text{simplify} \end{aligned}

So f(4)=17f(4) = 17. Likewise f(−2)=3(−2)+5=−6+5=−1f(-2) = 3(-2) + 5 = -6 + 5 = -1. Each input gives exactly one output.

Concept
Solving Literal Equations (Rearranging Formulas)

A literal equation has several letters, and we solve for one of them in terms of the others. The strategy is the same as solving a numerical equation: use inverse operations to isolate the wanted letter, treating every other letter as if it were a known number.

Example
Isolating a Variable in a Formula

The perimeter of a rectangle is P=2ℓ+2wP = 2\ell + 2w. Solve for the width ww.

P=2ℓ+2wstartP−2ℓ=2wsubtract 2ℓ from both sidesP−2ℓ2=wdivide both sides by 2\begin{aligned} P &= 2\ell + 2w &&\text{start}\\ P - 2\ell &= 2w &&\text{subtract } 2\ell \text{ from both sides}\\ \frac{P - 2\ell}{2} &= w &&\text{divide both sides by } 2 \end{aligned}

So w=P−2ℓ2w = \dfrac{P - 2\ell}{2}. Now any values of PP and ℓ\ell give ww directly, no re-solving needed.

Concept
Inequalities and the Sign-Flip Rule

An inequality uses <<, >>, ≤\le, or ≥\ge instead of an equals sign, and describes a whole range of values. You solve it almost exactly like an equation—add, subtract, multiply, and divide on both sides—with one crucial exception: whenever you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

Example
Solving an Inequality with a Flip

Solve −2x+1<9-2x + 1 < 9.

−2x+1<9start−2x<8subtract 1 from both sidesx>−4divide by −2: flip the sign\begin{aligned} -2x + 1 &< 9 &&\text{start}\\ -2x &< 8 &&\text{subtract } 1 \text{ from both sides}\\ x &> -4 &&\text{divide by } -2 \text{: \emph{flip} the sign} \end{aligned}

The solution is every number greater than −4-4. Check with x=0x = 0: −2(0)+1=1<9-2(0) + 1 = 1 < 9. ✓

Tip

Sign-flip trap: Adding or subtracting never changes the direction. Only multiplying or dividing by a negative flips << to >> (or ≥\ge to ≤\le). Dividing by a positive leaves it alone.

Concept
Factoring: The Distributive Property in Reverse

The distributive property a(b+c)=ab+aca(b + c) = ab + ac can be run backward. Factoring means pulling out a common factor shared by every term: ab+ac=a(b+c)ab + ac = a(b + c). To factor, find the greatest common factor (GCF) of all the terms, write it outside the parentheses, and put what remains inside.

Example
Expanding and Factoring as Opposites

Expanding removes parentheses; factoring restores them.

6(x+2)=6x+12⟷6x+12=6(x+2)6(x + 2) = 6x + 12 \qquad\longleftrightarrow\qquad 6x + 12 = 6(x + 2)

To factor 8y−208y - 20, the GCF of 88 and 2020 is 44, so 8y−20=4(2y−5)8y - 20 = 4(2y - 5). Check by re-distributing: 4(2y−5)=8y−204(2y - 5) = 8y - 20. ✓

Concept
Modeling Multi-Step Word Problems

Real problems become algebra in three moves: (1) let a letter stand for the unknown quantity; (2) translate each sentence into an expression or equation; (3) solve, then check the answer against the words. When two quantities are related, define both in terms of the same variable.

Example
Building and Solving a Model

A shirt costs $8\$8 more than a hat. Together they cost $34\$34. Find each price. Let hh be the hat's price; then the shirt costs h+8h + 8.

h+(h+8)=34total cost2h+8=34combine like terms2h=26subtract 8h=13divide by 2\begin{aligned} h + (h + 8) &= 34 &&\text{total cost}\\ 2h + 8 &= 34 &&\text{combine like terms}\\ 2h &= 26 &&\text{subtract } 8\\ h &= 13 &&\text{divide by } 2 \end{aligned}

The hat is $13\$13 and the shirt is 13+8=$2113 + 8 = \$21. Check: 13+21=3413 + 21 = 34. ✓

Concept
Turning a Pattern into a Formula

Many number patterns grow by the same amount each step. If a sequence starts at a first value and adds a fixed amount dd every term, the nnth term has a closed-form rule: (first term) + (n−1)⋅d+\ (n - 1)\cdot d. A closed form lets you jump straight to any term without listing them all.

Example
From Pattern to Closed Form
n = 1: 5n = 2: 8n = 3: 11n = 4: 14
Five starting squares, then three more at each step: 5+3(n−1)5 + 3(n-1).

The pattern 5,8,11,14,…5, 8, 11, 14, \dots starts at 55 and adds 33 each step. Its rule is

an=5+(n−1)⋅3=3n+2.a_n = 5 + (n - 1)\cdot 3 = 3n + 2.

Check: n=1n = 1 gives 3(1)+2=53(1) + 2 = 5; n=4n = 4 gives 3(4)+2=143(4) + 2 = 14. ✓ The 2020th term is 3(20)+2=623(20) + 2 = 62—found instantly, with no counting.

Tip

Big picture: A formula, a function, and a pattern's closed form are all the same idea—a compact rule that turns any input into its output. Learning to build these rules is what algebra is really about.

Why It Works

Expressions obey the same laws as numbers because a variable IS a number — just one we have not named. So x+x=2xx + x = 2x for the same reason 7+7=2⋅77 + 7 = 2\cdot7, and 2(x+3)=2x+62(x + 3) = 2x + 6 by the distributive property. Every algebraic move must remain true when a specific number is substituted for the variable, which is why substituting a value is a reliable check on any simplification.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.

Formulas, Proofs & Tips

Tip
Order of operations
Parentheses→Exponents→Multiply/Divide→Add/Subtract\text{Parentheses} \to \text{Exponents} \to \text{Multiply/Divide} \to \text{Add/Subtract}

What it means. A fixed order so every reader of an expression gets the same value.

Example. 3+4×2=3+8=113+4\times2=3+8=11 (multiply before adding), not 1414.

Why it works. Multiplication is repeated addition and exponents are repeated multiplication, so each level is a shorthand for the one below it and must be unpacked first. Parentheses override the order by grouping explicitly.

Tip. Multiply/divide are one level worked left to right, and so are add/subtract. 8−3+28-3+2 is 77, not 33.

Tip
The distributive property
a(b+c)=ab+aca(b+c)=ab+ac

What it means. Multiplying a sum multiplies each piece of it.

Example. 3(x+2)=3x+63(x+2)=3x+6.

Why it works. a(b+c)a(b+c) is b+cb+c added aa times. Regrouping those copies gives bb added aa times plus cc added aa times, i.e. ab+acab+ac. It is also the area of an a×(b+c)a\times(b+c) rectangle split into two.

Tip. Distribute the sign too: −(x−4)=−x+4-(x-4)=-x+4. Run it backwards to factor.