Examples in Special Products
SPECIAL PRODUCTS
(Examples)
• Products of the Sum and Difference of the Same Two terms
(x + y)(x − y) = x2 − y2, we have: 1. (s+2t)(s−2t)
= (s)2− (2t)2
= s2 − 4t2 2. (7s + 2t)(7s − 2t)
= (7s)2− (2t)2
= 49s2− 4t2 3. (12 + 5ab)(12 − 5ab)
= (12)2 − (5ab)2
= 144 − 25a2b2
• Square of a Sum of Binomials (x + y)2 = x2 + 2xy + y2, we have: 1. (5a + 2b)2
= (5a)2 + 2(5a)(2b) + (2b)2
= 25a2 + 20ab + 4b2 2. (3x + 10y)2
= (3x)2 + (2)(3x)(10y) + (10y)2
= 9x2+ 60xy + 100y2 3. (9z + 3)2
= (9z)2 + 2(9z)(3) + (3)2[Apply pattern.]
= 81z2 + 54z + 9[Simplify.]
• Square of a Difference of Binomials (x − y)2 = x2 − 2xy + y2, we have: 1. (3p − 4q)2
= (3p)2 − (2)(3p)(4q) + (4q)2
= 9p2 − 24pq + 16q2 2. (2a - 7)2
= (2a)2 - 2(2a)(7) + (7)2[Apply pattern.]
= 4a2 - 28a + 49[Simplify.] 3. (2x - 2)2
= (2x)2 - 2(2x)(2) + (2)2[Apply pattern.]
= 4x2 - 8x + 4[Simplify.] • Product of Two Binomials 1. (y + 4) (y + 2) = y2 + (4 + 2)y + 4(2) = y2 + 6y + 8 2. (6z + 3)( z + 7) = 6z2 + (7 + 6(3))z + 3(7) = 6z2 + 126z + 21 3. (x - 5) (x - 2) = x2 - (5 + 2)x – (5)2 = x2 - 7x + 10 4. (3x – 4y) (2x + 3y) = (3x) (2x) + (-4+ 2)xy – (4y) ( 3y) = 6x2 + xy – 12y2 • Square of Polynomials
|(x + 2 + 3y)2 | |
|= ([x + 2] + 3y)2 |[This is the (a + b)2 step.] |
|= [x + 2]2 + 2[x + 2](3y) + (3y)2 |[Here I apply (a + b)2 = a2 + 2ab + b2] |
|= [x2 + 4x + 4] + (2x + 4)(3y) + 9y2 |[In this row I just expand out the brackets.] |
|= x2 + 4x + 4 + 6xy +
(Examples)
• Products of the Sum and Difference of the Same Two terms
(x + y)(x − y) = x2 − y2, we have: 1. (s+2t)(s−2t)
= (s)2− (2t)2
= s2 − 4t2 2. (7s + 2t)(7s − 2t)
= (7s)2− (2t)2
= 49s2− 4t2 3. (12 + 5ab)(12 − 5ab)
= (12)2 − (5ab)2
= 144 − 25a2b2
• Square of a Sum of Binomials (x + y)2 = x2 + 2xy + y2, we have: 1. (5a + 2b)2
= (5a)2 + 2(5a)(2b) + (2b)2
= 25a2 + 20ab + 4b2 2. (3x + 10y)2
= (3x)2 + (2)(3x)(10y) + (10y)2
= 9x2+ 60xy + 100y2 3. (9z + 3)2
= (9z)2 + 2(9z)(3) + (3)2[Apply pattern.]
= 81z2 + 54z + 9[Simplify.]
• Square of a Difference of Binomials (x − y)2 = x2 − 2xy + y2, we have: 1. (3p − 4q)2
= (3p)2 − (2)(3p)(4q) + (4q)2
= 9p2 − 24pq + 16q2 2. (2a - 7)2
= (2a)2 - 2(2a)(7) + (7)2[Apply pattern.]
= 4a2 - 28a + 49[Simplify.] 3. (2x - 2)2
= (2x)2 - 2(2x)(2) + (2)2[Apply pattern.]
= 4x2 - 8x + 4[Simplify.] • Product of Two Binomials 1. (y + 4) (y + 2) = y2 + (4 + 2)y + 4(2) = y2 + 6y + 8 2. (6z + 3)( z + 7) = 6z2 + (7 + 6(3))z + 3(7) = 6z2 + 126z + 21 3. (x - 5) (x - 2) = x2 - (5 + 2)x – (5)2 = x2 - 7x + 10 4. (3x – 4y) (2x + 3y) = (3x) (2x) + (-4+ 2)xy – (4y) ( 3y) = 6x2 + xy – 12y2 • Square of Polynomials
|(x + 2 + 3y)2 | |
|= ([x + 2] + 3y)2 |[This is the (a + b)2 step.] |
|= [x + 2]2 + 2[x + 2](3y) + (3y)2 |[Here I apply (a + b)2 = a2 + 2ab + b2] |
|= [x2 + 4x + 4] + (2x + 4)(3y) + 9y2 |[In this row I just expand out the brackets.] |
|= x2 + 4x + 4 + 6xy +