HighSchoolQ

➕ Algebra 1 · Exponents and Polynomials

Special Products of Binomials

Learn the shortcuts for squaring a binomial and for (a + b)(a − b), why they work, why (a + b)² is not a² + b², and mental-math tricks, with practice.

Grades 9, 10⏱ 15 min✔ AI-checkedCCSS A-SSE.A.2CCSS A-APR.A.1CCSS A-APR.C.4
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-08.

  • A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
  • A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
  • A-APR.C.4: Prove polynomial identities and use them to describe numerical relationships.

By the end you can…

  • Expand (a + b)2, (a − b)2 and (a + b)(a − b) with the patterns a2 + 2ab + b2, a2 − 2ab + b2 and a2 − b2, where a and b can be any terms.
  • Explain why each pattern works, using an area picture and multiplying out, including why the middle term of a square is 2ab and why it disappears in (a + b)(a − b).
  • Show with numbers why (a + b)2 is not a2 + b2, and explain the difference between proving an identity and testing it with a few numbers.
  • Recognize which pattern fits (or that none does) and use the patterns for mental math.

Lesson (school style)

Some products of binomials come up so often that they have their own shortcuts. Squaring a binomial, (a + b)2, and multiplying a sum by a difference, (a + b)(a − b), always produce the same shapes. If you recognize the shape, you can write the answer without drawing a box. Every shortcut here comes from the multiplying you already know (every term times every term), so you can always rebuild it if you forget.

Squaring a sum

Squaring means multiplying by itself, so (a + b)2 = (a + b)(a + b). Multiply every term by every term:

(a + b)(a + b) = a × a + a × b + b × a + b × b = a2 + ab + ba + b2.

Since ba = ab, the two middle products combine into 2ab:

(a + b)2 = a2 + 2ab + b2.

In words: Square the first term, Double the product of the two terms, Square the last term (the S-D-S pattern). Here a and b can stand for any terms: numbers, letters, or products like 3x. For example, with a = x and b = 5: (x + 5)2 = x2 + 2(x)(5) + 52 = x2 + 10x + 25.

Why it works: the four-piece square

Picture a square with side a + b, for positive a and b. Cut it once each way, a from the corner. You get four pieces: an a-by-a square (area a2), a b-by-b square (area b2), and two a-by-b rectangles (area ab each). The whole square is the sum of its pieces: a2 + ab + ab + b2.

The two rectangles are the middle term, 2ab. It is sometimes called the cross term, because it comes from multiplying across the two different terms: a × b and b × a (the Outer and Inner products). These are not the cross products of a proportion.

The picture needs positive lengths, but the algebra works for every number. The multiplication above never assumed anything about a and b.

The classic mistake

It is tempting to square each piece and write (a + b)2 = a2 + b2. That treats the exponent as if it were handed out to each term. But an exponent on a sum means the whole sum is repeated as a factor.

A number test shows the problem. Try a = 3 and b = 4: (3 + 4)2 = 49, but 32 + 42 = 9 + 16 = 25. The 24 that went missing is exactly 2ab = 2 × 3 × 4, the two rectangles in the picture.

In fact, (a + b)2 = a2 + b2 only when 2ab = 0, that is, when a = 0 or b = 0. The same warning applies to differences: (a − b)2 is not a2 − b2.

Squaring a difference

Multiply every term by every term again:

(a − b)2 = (a − b)(a − b) = a2 − ab − ba + b2 = a2 − 2ab + b2.

When a and b are positive, the middle term is negative, because both cross products are negative. The last term is not negative, because (−b)(−b) = b2.

Another way to see it: a − b = a + (−b), so use the sum pattern with −b in place of b. That gives a2 + 2a(−b) + (−b)2. The minus sign in the middle comes from the 2a(−b).

A picture helps with the last term. Start with an a-by-a square and trim a strip b wide off two neighboring sides. Each strip has area ab, but the b-by-b corner lies in both strips, so subtracting both removes it twice. What is left is a square with side a − b. To get its area, subtract both strips and add b2 back once: (a − b)2 = a2 − ab − ab + b2. (This picture needs a > b > 0.)

Example: with a = 2x and b = 7, (2x − 7)2 = (2x)2 − 2(2x)(7) + 72 = 4x2 − 28x + 49. Notice that (2x)2 = 4x2: the power applies to the 2 as well as to the x.

Negative terms need care. The patterns are true for any numbers a and b, positive or negative. Take the first term as it is written, sign included. For (−2x + 5)2, a = −2x and b = 5, and the sum pattern gives (−2x)2 + 2(−2x)(5) + 52 = 4x2 − 20x + 25. The sign between the terms is a plus, but the middle term is negative, because the product of the two terms, (−2x)(5), is negative. In the same way, (−x − 3)2 = x2 + 6x + 9: the two signed terms are −x and −3, and their product is positive. A safe habit is to treat the binomial as a sum of two signed terms, such as (2x − 7) = (2x + (−7)), and double the product of those two signed terms. Or use the minus pattern with a positive b. Do not do both, because that counts the minus sign twice.

A sum times a difference

Now multiply a sum by a difference that use the same two terms:

(a + b)(a − b) = a2 − ab + ba − b2 = a2 − b2.

The middle terms −ab and +ba are opposites, so they cancel. There is no middle term, and the sign between the two squares is a minus. The order of the factors does not matter: (a − b)(a + b) gives the same answer.

For example, (x + 7)(x − 7) = x2 − 49. With a higher power, let a = x2 and b = 4: (x2 + 4)(x2 − 4) = (x2)2 − 42 = x4 − 16.

The pattern needs the same two terms, with a plus in one factor and a minus in the other. In the pattern, a is the term that has the same sign in both factors, and b is the term whose sign flips. For (x + 3)(3 − x), the 3 has the same sign in both and the x flips, so a = 3 and b = x. That gives 32 − x2 = 9 − x2, not x2 − 9. A cut-and-slide picture shows why it works. Take an a-by-a square and remove a b-by-b corner. Cut the L-shaped remainder into two pieces and slide them together into an (a + b)-by-(a − b) rectangle. (This picture needs a > b > 0.)

Which pattern is it?

Ask these questions in order:

  1. Is a binomial being squared, or are the two factors the same binomial? Use a square pattern. The middle term is 2ab, so (when the first term has no minus sign in front of it) it takes the sign between the two terms, and the last term is a square.
  2. Are the two factors the same two terms, with a plus in one and a minus in the other? Use a2 − b2, where a is the term whose sign stays the same and b is the term whose sign flips.
  3. Neither? Then there is no shortcut. Multiply every term by every term, as in Multiplying Polynomials.

For example, (x + 6)(x − 9) looks close to the second pattern, but 6 and 9 are different, so the middle terms do not cancel. It is an ordinary product: x2 − 9x + 6x − 54 = x2 − 3x − 54.

The patterns also run backward. A trinomial like x2 + 10x + 25 is a perfect square trinomial (the square of a binomial) when its first and last terms are squares (x2 = x × x and 25 = 5 × 5) and its middle term is twice the product of x and 5, with a plus or minus sign. So x2 + 10x + 25 = (x + 5)2. But x2 + 12x + 25 is not a binomial squared, because 2 × x × 5 = 10x, not 12x. The factoring lessons use this idea.

Mental math with the patterns

The same patterns work with plain numbers. Split a number into a friendly number plus or minus a small amount.

  • 212 = (20 + 1)2 = 202 + 2(20)(1) + 12 = 400 + 40 + 1 = 441.
  • 192 = (20 − 1)2 = 400 − 40 + 1 = 361.
  • 19 × 21 = (20 − 1)(20 + 1) = 202 − 12 = 399.

The product trick works when the two numbers are the same distance from a friendly middle number: one is d below it and the other is d above it. Then the product is the middle number squared minus d2. Any two numbers are the same distance from the number halfway between them, so the pattern is always true. It helps only when the halfway number is friendly. For 19 × 22 the halfway number is 20.5, so the trick is not much help.

Identities: proving and using them

An identity is an equation that is true for every value of its variables. (a + b)2 = a2 + 2ab + b2 is an identity, and multiplying it out with letters proves it, because the steps never depended on particular numbers.

Trying numbers is different. A number test can show that an equation is false, but a few matches do not prove it is true. For example, (a + b)2 = a2 + b2 + 24 is true when a = 3 and b = 4 (both sides are 49), but it fails when a = 1 and b = 1 (4 on the left, 26 on the right).

Identities also describe relationships between numbers. Subtracting the two square patterns gives (a + b)2 − (a − b)2 = 4ab. The numbers a + b and a − b have average a, and each is b away from it, so their squares differ by 4ab. The last worked example uses this.

Where the shortcuts stop

These patterns cover only a binomial squared and a sum times a difference of the same two terms. For anything else, such as (x + 3)(x − 5) or a binomial times a trinomial, go back to every term times every term.

Higher powers need care too. (a + b)3 is not a3 + b3: try a = 1 and b = 1, where (1 + 1)3 = 8 but 13 + 13 = 2. The cube of a binomial has its own pattern, which this lesson does not cover.

Key words

Binomial
A polynomial with two terms, like x + 5 or 2x − 7.
Square of a binomial
A binomial multiplied by itself, written (a + b)2 or (a − b)2. It means (a + b)(a + b), not a2 + b2.
Perfect square trinomial
The trinomial you get from squaring a binomial: a2 + 2ab + b2 or a2 − 2ab + b2, such as x2 + 10x + 25.
Difference of squares
An expression of the form a2 − b2: one square minus another. It is the answer to (a + b)(a − b).
Cross term
The middle term of a binomial squared, 2ab or −2ab. It comes from multiplying across the two different terms, a × b and b × a (the Outer and Inner products). It has nothing to do with the cross products of a proportion.
Identity
An equation that is true for every value of its variables, like (a + b)2 = a2 + 2ab + b2. Multiplying out with letters proves it.

Worked examples

Example 1. Expand (x + 5)2.

  1. This is a binomial squared with a plus sign, so use the square-of-a-sum pattern with a = x and b = 5.
  2. Square the first term: x2.
  3. Double the product of the two terms: 2 × x × 5 = 10x.
  4. Square the last term: 52 = 25.
  5. Put the three parts together: x2 + 10x + 25.
  6. Check with x = 3: (3 + 5)2 = 64, and 9 + 30 + 25 = 64. The wrong answer x2 + 25 would give 34.

Answer: Answer: x2 + 10x + 25

Example 2. Expand (2x − 7)2.

  1. This is a binomial with a minus sign, squared. Use (a − b)2 = a2 − 2ab + b2 with a = 2x and b = 7.
  2. Square the first term, including the 2: (2x)2 = 4x2.
  3. Double the product, keeping the minus sign: −2 × 2x × 7 = −28x.
  4. Square the last term. It stays positive: 72 = 49.
  5. The answer is 4x2 − 28x + 49.
  6. Check with x = 5: (10 − 7)2 = 9, and 4(25) − 28(5) + 49 = 100 − 140 + 49 = 9.

Answer: Answer: 4x2 − 28x + 49

Example 3. Expand (3x + 2y)2.

  1. Here a = 3x and b = 2y. Both parts are products, so square every factor in them.
  2. a2 = (3x)2 = 9x2.
  3. 2ab = 2 × 3x × 2y = 12xy.
  4. b2 = (2y)2 = 4y2.
  5. The answer is 9x2 + 12xy + 4y2.
  6. Check with x = 3 and y = 4: (9 + 8)2 = 289, and 81 + 144 + 64 = 289.

Answer: Answer: 9x2 + 12xy + 4y2

Example 4. Expand (−4x + 3)2.

  1. The first term is −4x, and it keeps its sign. So a = −4x and b = 3. The sign between the terms is a plus, so use a2 + 2ab + b2.
  2. Square the first term: (−4x)2 = 16x2. The square of a negative is positive.
  3. Double the product of the two terms: 2 × (−4x) × 3 = −24x.
  4. Square the last term: 32 = 9.
  5. The answer is 16x2 − 24x + 9. The sign between the terms was a plus, but the middle term is negative, because (−4x)(3) is negative.
  6. Cross-check: −4x + 3 is the opposite of 4x − 3, and an opposite has the same square, so (−4x + 3)2 = (4x − 3)2 = 16x2 − 24x + 9 by the minus pattern.
  7. Check with x = 4: (−16 + 3)2 = 169, and 256 − 96 + 9 = 169.

Answer: Answer: 16x2 − 24x + 9

Example 5. Expand (x + 7)(x − 7).

  1. The two factors use the same two terms, x and 7, with a plus in one and a minus in the other. Use (a + b)(a − b) = a2 − b2 with a = x and b = 7.
  2. Square the first term: x2. Square the last term: 72 = 49. Subtract: x2 − 49.
  3. There is no middle term. The full product is x2 − 7x + 7x − 49, and the −7x and +7x cancel.
  4. Check with x = 3: (3 + 7)(3 − 7) = 10 × (−4) = −40, and 9 − 49 = −40.

Answer: Answer: x2 − 49

Example 6. Expand (x2 + 4)(x2 − 4).

  1. These are the same two terms, x2 and 4, with opposite signs between them. So a = x2 and b = 4.
  2. a2 = (x2)2 = x4 (a power of a power: multiply the exponents).
  3. b2 = 42 = 16.
  4. The answer is x4 − 16.
  5. Check with x = 3: (9 + 4)(9 − 4) = 13 × 5 = 65, and 81 − 16 = 65.

Answer: Answer: x4 − 16

Example 7. Does a shortcut apply to (x + 6)(x − 9)? Expand it.

  1. The two factors are not the same binomial, so this is not a square.
  2. They do not use the same two terms (6 versus 9), so it is not a sum times a difference. No shortcut applies.
  3. Multiply every term by every term: x × x = x2, x × (−9) = −9x, 6 × x = 6x and 6 × (−9) = −54.
  4. Combine the like terms: −9x + 6x = −3x. The answer is x2 − 3x − 54.
  5. Check with x = 3: (9)(−6) = −54, and 9 − 9 − 54 = −54. Forcing the difference-of-squares pattern would give x2 − 54, which is −45 at x = 3.

Answer: Answer: No shortcut applies. (x + 6)(x − 9) = x2 − 3x − 54

Example 8. Use special products to find 212 and 19 × 21 in your head.

  1. Split 21 into 20 + 1 and use the square-of-a-sum pattern: 212 = 202 + 2(20)(1) + 12.
  2. 400 + 40 + 1 = 441.
  3. For 19 × 21, notice that 19 and 21 are each 1 away from 20: 19 × 21 = (20 − 1)(20 + 1) = 202 − 12.
  4. 400 − 1 = 399.
  5. Check by ordinary multiplication: 21 × 21 = 441, and 19 × 21 = 420 − 21 = 399.

Answer: Answer: 212 = 441 and 19 × 21 = 399

Example 9. Prove that (a + b)2 − (a − b)2 = 4ab, then use it to find 342 − 262 without squaring either number.

  1. Expand each square: (a + b)2 = a2 + 2ab + b2 and (a − b)2 = a2 − 2ab + b2.
  2. Subtract, keeping the second square in parentheses so its signs flip: (a2 + 2ab + b2) − (a2 − 2ab + b2) = a2 + 2ab + b2 − a2 + 2ab − b2.
  3. The a2 terms cancel, the b2 terms cancel, and 2ab + 2ab = 4ab. No particular numbers were used, so this is true for every a and b.
  4. Match 34 = a + b and 26 = a − b. Their average is a = 30, and each is b = 4 away from it.
  5. So 342 − 262 = 4ab = 4 × 30 × 4 = 480.
  6. Check by squaring: 342 = 1156 and 262 = 676, and 1156 − 676 = 480.

Answer: Answer: The identity is proved, and 342 − 262 = 480

In plain English

Imagine a square patio that is 7 feet on each side. Its area is 7 × 7 = 49 square feet.

Now split each side into a 3-foot part and a 4-foot part. The patio breaks into four pieces: a 3 by 3 square (9), a 4 by 4 square (16), and two 3 by 4 rectangles (12 each).

Add them up: 9 + 16 + 12 + 12 = 49. The two rectangles are the middle of the pattern. If you forget them, you get 9 + 16 = 25, and 24 square feet go missing.

That is why (a + b)2 is a2 + 2ab + b2 and not just a2 + b2. The 2ab is the two rectangles.

There is one more pattern. A sum times a difference, like (a + b)(a − b), has two middle pieces that cancel. You get a2 − b2.

See it

A square with side a + b, cut into four pieces a b a b a2 ab ab b2 (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 Common mistake (a + b)2 ≠ a2 + b2 The two ab pieces go missing. Drawn for positive a and b, and not to scale. The identity holds for every number.
The square of side a + b has four pieces: a2, b2 and two ab rectangles. Leave out the rectangles and you get the classic mistake.
Why (a + b)(a − b) = a2 − b2: cut and slide Square minus a corner Same pieces, rearranged B A b2 a − b b b a − b a turn and slide B A B a + b a − b a2 − b2 the square minus the b2 corner (a + b)(a − b) the same pieces in a new shape The areas match: (a + b)(a − b) = a2 − b2 Drawn for a > b > 0 and not to scale. The identity holds for every number.
Cut the L-shape into two pieces and slide them into a rectangle: a2 − b2 = (a + b)(a − b).

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Food

The cookie tray

A bakery sells cookies on a square tray with 11 rows of 11 cookies. Counting row by row is slow. Instead, see the tray as a 10-by-10 block of 100 cookies, plus a strip of 10 along one side, plus a strip of 10 along the other side, plus 1 cookie in the corner.

How it connects: (10 + 1)2 = 102 + 2(10)(1) + 12 = 100 + 20 + 1 = 121. The two strips of 10 are the middle term 2ab, and the corner cookie is b2. Someone who writes 112 = 102 + 12 gets 101 and loses the 20 cookies in the strips. The limit: the picture needs positive counts, but the pattern works for every number.

Sports

Trimming a gym mat

A square gym mat is 12 feet on each side. The coach trims 2 feet off two neighboring sides, so the mat is now 10 feet by 10 feet. Each trimmed strip is 12 feet by 2 feet, which is 24 square feet. But the 2-by-2 corner belongs to both strips, so the total removed is 24 + 24 − 4 = 44 square feet.

How it connects: (12 − 2)2 = 122 − 2(12)(2) + 22 = 144 − 48 + 4 = 100. Subtracting both strips removes the corner twice, so the +b2 gives it back once. That is why the last term of (a − b)2 is positive. The limit: the picture needs a > b > 0, but the pattern holds for every number.

Gaming

Stretch and squash

In a game’s level editor, a square tile is 20 pixels wide and 20 pixels tall. You stretch it 3 pixels wider and squash it 3 pixels shorter, so it is 23 by 17. It feels like you added as much as you took away, but the area went from 20 × 20 = 400 to 23 × 17 = 391, which is 9 less.

How it connects: (20 + 3)(20 − 3) = 202 − 32 = 391. The Outer and Inner products, −60 and +60, cancel, and only −b2 is left. The limit: this works because the same amount, 3, is added and removed. Stretching by 3 and squashing by 2 gives (20 + 3)(20 − 2) = 414, which needs a middle term.

Proverb

Haste makes waste

This saying means that rushing a job usually leads to mistakes you have to redo, so you waste time instead of saving it.

How it connects: Writing (a + b)2 = a2 + b2 in a hurry feels fast, but the answer is wrong. Take one extra line to write the repeated factor, (a + b)(a + b), and multiply every term by every term. The limit: the lesson is to slow down at that one step, not to multiply out everything forever. Once you know the pattern, (a + b)2 = a2 + 2ab + b2 is both fast and correct.

Idiom

Cancel each other out

When two things “cancel each other out,” they have opposite effects of the same size, so nothing is left. Gaining 5 points and then losing 5 points cancel each other out.

How it connects: In (a + b)(a − b) the outer and inner products are −ab and +ab. They cancel each other out, so there is no middle term: a2 − b2. The limit: they cancel only when the two factors use the same two terms. In (a + b)(a − c) the middle terms are −ac and +ab, which do not cancel unless b = c (or a = 0).

Everyday joke

The cross term

Why is the cross term always so cross? Everybody squaring a + b keeps leaving it out.

How it connects: The middle term 2ab is called the cross term because it comes from multiplying across the two different terms, a × b and b × a. Writing (a + b)2 = a2 + b2 leaves it out. The limit: in (a + b)(a − b) the Outer and Inner products really do cancel, so there is no cross term to put back in. Do not add one there.

Memory help

Acronym

S-D-S

For a binomial multiplied by itself: Square the first term, Double the product of the two terms, Square the last term. Use each term with its own sign. For (3m − 2)2 the terms are 3m and −2: square the first to get 9m2, double the product 2(3m)(−2) = −12m, and square the last, (−2)2 = 4. The answer is 9m2 − 12m + 4. S-D-S fits only a binomial squared. It does not fit (a + b)(a − b).

Rhyme

Plus-minus twins

Same two terms with a plus and a minus: the middle cancels, there is no fuss. Square the term that keeps its sign, then subtract the square of the one that flips. Two terms are left, and it is over fast. (In (x + 3)(3 − x) the 3 keeps its sign, so the answer is 9 − x2.)

Picture it

The cut square

Draw a square, cut it once each way, and label the four pieces a2, ab, ab and b2. The two squares sit on one diagonal and the two matching rectangles on the other. If your answer shows only the two squares, the rectangles are missing.

Trap alert

The missing rectangles

(n + 14)2 is not n2 + 196. The two rectangles add 28n, so (n + 14)2 = n2 + 28n + 196. At n = 3 the left side is 289, but n2 + 196 gives only 205.

Trap alert

A minus in the middle, a plus at the end

(c − 7)2 = c2 − 14c + 49. Here the middle term takes the minus sign from the binomial, but the last term is (−7)2 = +49. It is not c2 − 49 (that is the answer to (c + 7)(c − 7)), and it is not c2 − 14c − 49.

Sketch it

Cut and slide

Sketch a square with a small square missing from one corner. Cut the L-shape into two rectangles, turn one, and slide them together into one long rectangle. That is a2 − b2 becoming (a + b)(a − b).

Study notes

Key ideas

  • (a + b)2 = a2 + 2ab + b2 and (a − b)2 = a2 − 2ab + b2. Square the first term, double the product, square the last term. When the first term has no minus sign in front of it, the middle term takes the sign between the two terms. The last term b2 is never negative.
  • The patterns hold for negative a and b too. A first term with a minus sign in front of it can make the middle term negative even when the sign between the terms is a plus: (−2x + 5)2 = 4x2 − 20x + 25. Treat the binomial as a sum of two signed terms and double the product of those terms.
  • (a + b)(a − b) = a2 − b2: the middle terms cancel. It needs the same two terms in both factors, with opposite signs between them.
  • (a + b)2 ≠ a2 + b2 unless a = 0 or b = 0. The missing piece is the cross term 2ab.
  • a and b can be any terms, like 3x, x2 or 2y. Square the whole term: (3x)2 = 9x2.
  • These are identities, true for every value. Multiplying out with letters proves them. Testing a few numbers does not.

Steps

  1. Decide which pattern fits: a binomial squared, a sum times a difference of the same two terms, or neither.
  2. Name a and b. Take a as the first term exactly as written, including its sign. Take b as the second term without the sign between the two terms. If the sign between them is a plus, use a2 + 2ab + b2. If it is a minus, use a2 − 2ab + b2. For a sum times a difference, a is the term whose sign stays the same and b is the term whose sign flips, and the answer is a2 − b2.
  3. Do not count a minus sign twice. Either use the minus pattern with a positive b, or write a − b as a + (−b) and use the plus pattern with a negative b. Examples: (2x − 7)2 has a = 2x and b = 7, and (−2x + 5)2 has a = −2x and b = 5.
  4. Write the pattern with a and b in parentheses, then simplify each part completely: (3x)2 = 9x2.
  5. Check with a test value such as x = 3.

Formulas

  • Square of a sum: (a + b)2 = a2 + 2ab + b2
  • Square of a difference: (a − b)2 = a2 − 2ab + b2
  • Sum times difference: (a + b)(a − b) = a2 − b2
  • Squares that differ: (a + b)2 − (a − b)2 = 4ab
  • Mental-math product: (m − d)(m + d) = m2 − d2

Watch out

  • (a + b)2 ≠ a2 + b2. Write the repeated factor first: (a + b)(a + b).
  • (a − b)2 ≠ a2 − b2. The expression a2 − b2 is the answer to (a + b)(a − b).
  • The last term of (a − b)2 is +b2, not −b2, because (−b)(−b) = b2.
  • Square the whole term: (3x)2 = 9x2, not 3x2.
  • A shortcut needs its exact shape. (x + 3)(x − 5) is neither a square nor a sum times a difference of the same two terms.
  • A few matching numbers do not prove an identity.

Why it works

  • Why 2ab? Multiplying (a + b)(a + b) gives two middle products, ab and ba (the Outer and Inner products). They are equal, so they combine into 2ab. In the picture they are the two rectangles.
  • Why no middle term in (a + b)(a − b)? The Outer and Inner products are −ab and +ab, which cancel.
  • Why learn the patterns? They save time and make mental math easy. They also help you see structure: x4 − 16 is (x2)2 − 42, a difference of squares. Read backward, the patterns become the factoring shortcuts you will use in the factoring unit.

Common mistakes

✗ Mistake(a + b)2 = a2 + b2 (squaring each piece).
✓ InsteadSquaring multiplies the whole sum by itself: (a + b)(a + b) = a2 + 2ab + b2. The cross term 2ab is part of the answer. For a = 2 and b = 3, (2 + 3)2 = 25 but 22 + 32 = 13.
✗ Mistake(x − 13)2 = x2 − 169 (the square of a difference is a difference of squares).
✓ Insteadx2 − 169 is the answer to (x + 13)(x − 13). Squaring the difference gives (x − 13)2 = x2 − 26x + 169.
✗ Mistake(x − 13)2 = x2 − 26x − 169 (the last term keeps the minus sign).
✓ InsteadThe last term is (−13)2 = +169. Only the middle term is negative: x2 − 26x + 169.
✗ MistakeIn (−2x + 5)2 the sign between the terms is a plus, so every term of the answer is positive.
✓ InsteadThe first term is −2x, so the double product is 2 × (−2x) × 5 = −20x. The answer is 4x2 − 20x + 25.
✗ Mistake(a + b)(a − b) is the same thing as (a − b)2, so it has a middle term.
✓ InsteadThey are different. (a − b)2 = a2 − 2ab + b2 has a middle term, but in (a + b)(a − b) the middle terms −ab and +ab cancel, giving a2 − b2.
✗ Mistake(3x + 2)2 = 3x2 + 12x + 4 (not squaring the coefficient).
✓ InsteadSquare the whole first term: (3x)2 = 9x2. So (3x + 2)2 = 9x2 + 12x + 4.
✗ Mistake(x + 2)(x − 9) is a difference of squares, so it equals x2 − 18.
✓ InsteadThe pattern needs the same two terms in both factors. Here 2 and 9 differ, so the middle terms do not cancel: (x + 2)(x − 9) = x2 − 7x − 18.

Teach it back

Explain to a friend (or a pet!) why (n + 6)2 is not n2 + 36, how to expand it, and what the picture of the square shows.

Explaining it out loud shows what you really know. Did you…

  • I wrote (n + 6)2 = (n + 6)(n + 6) and multiplied every term by every term.
  • I got n2 + 12n + 36 and said where the 12n comes from: 2 × n × 6, the two rectangles in the picture.
  • I showed with n = 3 that (3 + 6)2 = 81 but 32 + 36 = 45, so n2 + 36 cannot be right.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Multiplying PolynomialsDifference of SquaresFactoring Trinomials