Notesale: Turn your study into money

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Polynomial Special Products

 Product of Sum and Difference

ሺܽ + ܾሻሺܽ − ܾሻ = ܽଶ − ܾ ଶ

 equal to the square of “a” minus the square of “b”

Example:

ሺ5‫ ݏ‬ସ ‫ ݐ‬ଷ + 11‫ ݔ‬ସ ‫ ݕ‬ଷ ሻሺ5‫ ݏ‬ସ ‫ ݐ‬ଷ − 11‫ ݔ‬ସ ‫ ݕ‬ଷ ሻ
⤷ ܽ = 5‫ ݏ‬ସ ‫ ݐ‬ଷ ; ܾ = 11‫ ݔ‬ସ ‫ ݕ‬ଷ

⤷ ܽଶ − ܾ ଶ

= ሺ5‫ ݏ‬ସ ‫ ݐ‬ଷ ሻଶ − ሺ11‫ ݔ‬ସ ‫ ݕ‬ଷ ሻଶ

= ૛૞࢙ૡ ࢚૟ − ૚૛૚࢞ૡ ࢟૟

‫ݔ‬ଶ‫ݕ‬ଷ
3‫଺ ݎ‬
‫ݔ‬ଶ‫ݕ‬ଷ
3‫଺ ݎ‬
ቆ ସ + ସ ଺ቇ ቆ ସ − ସ ଺ቇ
2‫ݖ‬
6‫ݐ ݏ‬
2‫ݖ‬
6‫ݐ ݏ‬
⤷ܽ=

‫ݔ‬ଶ‫ݕ‬ଷ
3‫଺ ݎ‬
;
ܾ
=
2‫ ݖ‬ସ
6‫ ݏ‬ସ ‫଺ ݐ‬

⤷ ܽଶ − ܾ ଶ

ଶ

ଶ

3‫଺ ݎ‬
‫ݔ‬ଶ‫ݕ‬ଷ
= ቆ ସ ቇ − ቆ ସ ଺ቇ
2‫ݖ‬
6‫ݐ ݏ‬
=

࢞ ૝ ࢟૟
ૢ࢘૚૛
−
૝ࢠૡ ૜૟࢙ૡ ࢚૚૛

ሺ106ሻሺ94ሻ

⤷ 106 = 100 + 6 ; 94 = 100 − 6
⤷ ܽ = 100 ; ܾ = 6
⤷ ܽଶ − ܾ ଶ

= ሺ100ሻଶ − ሺ6ሻଶ

= 10000 − 36
= ૢૢ૟૝

 Square of Binomial

ሺܽ + ܾሻଶ = ܽଶ + 2ܾܽ + ܾ ଶ

 first term is “+” square of “a”
 middle term is “+” and equal to the twice the product of “a” and “b”
 last term is “+” square of “b”

ሺܽ − ܾሻଶ = ܽଶ − 2ܾܽ + ܾ ଶ

 first term is “+” square of “a”
 middle term is “−” and equal to the twice the product of “a” and “b”
 last term is “+” square of “b”

Example:

ሺ2‫ ݏ‬ସ ‫ ݐ‬ଷ + 3‫ ݔ‬ସ ‫ ݕ‬ଷ ሻଶ

⤷ ܽ = 2‫ ݏ‬ସ ‫ ݐ‬ଷ ; ܾ = 3‫ ݔ‬ସ ‫ ݕ‬ଷ

⤷ ܽଶ + 2ܾܽ + ܾ ଶ

= ሺ2‫ ݏ‬ସ ‫ ݐ‬ଷ ሻଶ + 2ሺ2‫ ݏ‬ସ ‫ ݐ‬ଷ ሻሺ3‫ ݔ‬ସ ‫ ݕ‬ଷ ሻ + ሺ3‫ ݔ‬ସ ‫ ݕ‬ଷ ሻଶ

= ૝࢙ૡ ࢚૟ + ૚૛࢙૝ ࢚૜ ࢞૝ ࢟૜ + ૢ࢞ૡ ࢟૟
ሺ2‫ ݏ‬ସ ‫ ݐ‬ଷ − 3‫ ݔ‬ସ ‫ ݕ‬ଷ ሻଶ

⤷ ܽ = 2‫ ݏ‬ସ ‫ ݐ‬ଷ ; ܾ = 3‫ ݔ‬ସ ‫ ݕ‬ଷ

⤷ ܽଶ − 2ܾܽ + ܾ ଶ

= ሺ2‫ ݏ‬ସ ‫ ݐ‬ଷ ሻଶ − 2ሺ2‫ ݏ‬ସ ‫ ݐ‬ଷ ሻሺ3‫ ݔ‬ସ ‫ ݕ‬ଷ ሻ + ሺ3‫ ݔ‬ସ ‫ ݕ‬ଷ ሻଶ

= ૝࢙ૡ ࢚૟ − ૚૛࢙૝ ࢚૜ ࢞૝ ࢟૜ + ૢ࢞ૡ ࢟૟
ଶ

3‫ ݕݔ‬ଶ
2‫ ݎ‬ଷ
ቆ ଷ + ଶ ସቇ
‫ݖ‬
5‫ݐ ݏ‬
⤷ܽ=

3‫ ݕݔ‬ଶ
2‫ ݎ‬ଷ
;
ܾ
=
‫ݖ‬ଷ
5‫ ݏ‬ଶ ‫ ݐ‬ସ

⤷ ܽଶ + 2ܾܽ + ܾ ଶ
ଶ

ଶ

3‫ ݕݔ‬ଶ
3‫ ݕݔ‬ଶ
2‫ ݎ‬ଷ
2‫ ݎ‬ଷ
= ቆ ଷ ቇ + 2 ቆ ଷ ቇ ቆ ଶ ସቇ + ቆ ଶ ସቇ
‫ݖ‬
‫ݖ‬
5‫ݐ ݏ‬
5‫ݐ ݏ‬

ૢ࢞૛ ࢟૝ ૚૛࢞࢟૛ ࢘૜
૝࢘૟
=
+
+
ࢠ૟
૞ࢠ૜ ࢙૛ ࢚૝ ૛૞࢙૝ ࢚ૡ

 Cube of Binomial

ሺܽ + ܾሻଷ = ܽଷ + 3ܽଶ ܾ + 3ܾܽ ଶ + ܾ ଷ





first term is “+” cube of “a”
second term is “+” and equal to the product of 3, square of “a” and “b”
third term is “+” and equal to the product of 3, “a” and square of “b”
last term is “+” cube of “b”






first term is “+” cube of “a”
second term is “−” and equal to the product of 3, square of “a” and “b”
third term is “+” and equal to the product of 3, “a” and square of “b”
last term is “−” cube of “b”

ሺܽ − ܾሻଷ = ܽଷ − 3ܽଶ ܾ + 3ܾܽ ଶ − ܾ ଷ

Example:

ሺ4݉ଶ + 5݊ଷ ሻଷ

⤷ ܽ = 4݉ଶ ; ܾ = 5݊ଷ

⤷ ܽଷ + 3ܽଶ ܾ + 3ܾܽ ଶ + ܾ ଷ

= ሺ4݉ଶ ሻଷ + 3ሺ4݉ଶ ሻଶ ሺ5݊ଷ ሻ + 3ሺ4݉ଶ ሻሺ5݊ଷ ሻଶ + ሺ5݊ଷ ሻଷ
= ૟૝࢓૟ + ૛૝૙࢓૝ ࢔૜ + ૜૙૙࢓૛ ࢔૟ + ૚૛૞࢔ૢ

ሺ4݉ଶ − 5݊ଷ ሻଷ

⤷ ܽ = 4݉ଶ ; ܾ = 5݊ଷ

⤷ ܽଷ − 3ܽଶ ܾ + 3ܾܽ ଶ − ܾ ଷ

= ሺ4݉ଶ ሻଷ − 3ሺ4݉ଶ ሻଶ ሺ5݊ଷ ሻ + 3ሺ4݉ଶ ሻሺ5݊ଷ ሻଶ − ሺ5݊ଷ ሻଷ
= ૟૝࢓૟ − ૛૝૙࢓૝ ࢔૜ + ૜૙૙࢓૛ ࢔૟ − ૚૛૞࢔ૢ

Additional Notes:

 the exponent of “a” is decreasing from left to right

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