Summary Algebra Polynomials Special Products

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
‫ݔ‬ଶ‫ݕ‬ଷ 3‫଺ ݎ‬
= 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