Multiplying polynomials needs nothing more than the distributive law, but a few common forms are much faster once remembered as product formulas.
Cutting a square of side a + b into the four pieces a², ab, ab and b² shows (a + b)² = a² + 2ab + b² directly as area.
Changing the sign gives (a − b)², where only the middle term becomes −2ab, and in the product of a sum and a difference, (a + b)(a − b), the middle terms cancel and only a² − b² is left.
These formulas turn calculations such as 101² or 51 × 49 into mental arithmetic, and in the next topic they are used in reverse for factorising.
(a + b)² as area
seegongsik.com
An area picture of a square of side a + b cut into the four pieces a², ab, ab and b²
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📐 What the area picture shows
①The area of a square of side a + b is (a + b)².
②This square splits into four pieces: a², ab, ab and b².
③So (a + b)² = a² + 2ab + b².
④Writing (a + b)² as a² + b² leaves out the two ab pieces. With a = 3 and b = 2 in the picture, 25 and 13 differ by 12.
The square of a difference (a − b)²
Square of a sum
(a + b)² = a² + 2ab + b²
Add the squares of the two numbers and twice their product.
Square of a difference
(a − b)² = a² − 2ab + b²
Only the sign of the middle term changes.
💡 Getting the signs right
①The middle term of (a + b)² is +2ab.
②The middle term of (a − b)² is −2ab.
③The end terms a² and b² are squares, so they are always positive.
④(x + a)(x + b) = x² + (a + b)x + ab is another expansion formula used alongside them.
Sum times difference (a + b)(a − b)
seegongsik.com
The L-shape left after removing b² from a square of side a, cut and rejoined into a rectangle of width a + b and height a − b
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Sum times difference
(a + b)(a − b) = a² − b²
When expanded, +ab and −ab cancel each other.
Using the product formulas
Mental arithmetic
101² = (100 + 1)² = 10000 + 200 + 1 = 10201
Splitting a number into easy parts and using a product formula makes large products easy to do in your head.
🧮 Examples
①99² = (100 − 1)² = 10000 − 200 + 1 = 9801.
②51 × 49 = (50 + 1)(50 − 1) = 2500 − 1 = 2499.
③(√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3: the formulas work just the same with square roots.
Worked examples
Example 1
Expand (x + 3)².
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Put a = x and b = 3 into the square-of-a-sum formula (a + b)² = a² + 2ab + b².
(x + 3)² = x² + 2·x·3 + 3²
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Work out each term and simplify.
= x² + 6x + 9
▸ x² + 6x + 9
Do not leave out the middle term 2ab of a perfect square.
Example 2
Expand (x + 2)(x − 5).
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Put a = 2 and b = −5 into (x + a)(x + b) = x² + (a + b)x + ab.
x² + (2 + (−5))x + 2·(−5)
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Work out the sum 2 + (−5) = −3 and the product 2 × (−5) = −10.
= x² − 3x − 10
▸ x² − 3x − 10
The coefficient of x is the sum of the two numbers, and the constant term is their product.
Exam checklist
Product formulas
(a ± b)² = a² ± 2ab + b², (a + b)(a − b) = a² − b²
These formulas are the foundation for factorising and quadratic equations.
Grade-9 school exam type
Expand (x + 4)(x − 4).
①x² − 8
②x² + 16
③x² − 16
④x² − 8x − 16
⑤x² − 16x
▸ ③ x² − 16
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Use the sum-and-difference product (a + b)(a − b) = a² − b².
(x + 4)(x − 4) = x² − 4²
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Compute.
= x² − 16
🎯 Exam points
①(a + b)² is not a² + b²; do not leave out 2ab.
②The sum-times-difference formula appears even more often in reverse, as factorising.
③The formulas apply unchanged to expressions with square roots or fractions.
④Learn (x + a)(x + b) = x² + (a + b)x + ab as well.
⑤An answer is finished only after like terms have been collected.