Special Product & Expansion –
We can find the products of certain types of algebraic expressions directly without actually carrying out the multiplication. These products called special products, come in handy while simplifying expressions or solving equations. Some of the special products are as follows –
A) (x+a).(x+b) = x²+ (a+b)x + ab
Proof - (x+a).(x+b) = (x+a).x + (x+a).b
= x² + ax + bx + ab = x² + (a+b)x + ab (proven)
There are some example are given below for your better understanding –
Example.1) (x+5)(x+4) = x² + (5+4)x + 5.4
[applying formula (x+a).(x+b) = x²+ (a+b)x + ab]
= x² + 9x + 20 (Ans.)
Example.2) (x+3)(x+7) = x² + (7+3)x + 3.7
[applying formula (x+a).(x+b) = x² + (a+b)x + ab]
= x² + 10x + 21 (Ans.)
Example.3) (3a²+4bᶟ)(5a²+3bᶟ) = 3.5(a²)² + (4.5+3.3)a²bᶟ+ 4.3(bᶟ)² [applying formula (x+a).(x+b) = x² + (a+b)x + ab]
= 15a²⁺² + (20+9)a²bᶟ + 12bᶟ⁺ᶟ
= 15a⁴ + 29 a²bᶟ + 12b⁶ (Ans.)
Example.4) (2ab+4xy)(3ab+5xy) = 2.3(ab)²+ (4.3+2.5)ab.xy + 4.5(xy)² [applying formula (x+a).(x+b) = x² + (a+b)x + ab]
= 6a²b² + (12+10)abxy + 20x²y²
= 6a²b²+ 22abxy + 20x²y² (Ans.)
B) (x+a).(x-b) = x²+ (a-b).x – ab
Proof - (x+a).(x-b) = (x+a).x – (x+a).b
= x² + ax – bx – ab
= x² + (a-b).x – ab (Proven)
There are some examples are given below for your better understanding –
Example.1) (x+8)(x-5) = x²+ (8-5)x – 8.5
[applying formula (x+a).(x-b) = x² + (a-b).x – ab]
= x² + 3x – 40 (Ans.)
Example.2) (a+5)(a-6) = a² + (5-6)a – 5.6
[applying formula (x+a).(x-b) = x² + (a-b).x – ab]
= a² + (-1)a – 30 = a²- a – 30 (Ans.)
Example.3) (2x+3a)(3x-4a) = 2.3x²+ (3.3-2.4)ax – 3.4a²
[applying formula (x+a).(x-b) = x² + (a-b).x – ab]
= 6x²+ (9-8)ax – 12a² = 6x² + ax – 12a² (Ans.)
Example.4) (5ab+2xy)(4ab-5xy) = 5.4.(ab)²+ (2.4-5.5)ab.xy – 2.5.(xy)² [applying formula (x+a).(x-b) = x² + (a-b).x – ab]
= 20a²b² + (8-25)abxy – 10x²y²
= 20a²b²- 17abxy - 10x²y² (Ans.)
C) (x-a)(x+b) = x²- (a-b).x - ab
Proof - (x-a).(x+b) = (x-a).x + (x-a).b
= x² -ax + bx –ab
= x² - (a-b)x – ab (proven)
Example.1) (x-8)(x+5) = x²- (8-5)x – 8.5
[applying formula (x-a).(x+b) = x²- (a-b).x – ab]
= x² - 3x – 40 (Ans.)
Example.2) (a-6)(a+5) = a²- (6-5)a – 6.5
[applying formula (x-a).(x+b) = x² - (a-b).x – ab]
= a²- (1)a – 30 = a²- a – 30
Example.3) (3a²-4bᶟ)(5a²+3bᶟ) = 3.5(a²)²- (4.5-3.3)a²bᶟ- 4.3(bᶟ)² [applying formula (x-a).(x+b) = x² - (a-b).x - ab]
= 15a²⁺²- (20+9)a²bᶟ - 12bᶟ⁺ᶟ
= 15a⁴ - 29 a²bᶟ - 12b⁶ (Ans.)
Example.4) (2ab-4xy)(3ab+5xy) = 2.3(ab)²- (4.3-2.5)ab.xy - 4.5(xy)² [applying formula (x-a).(x+b) = x² - (a-b)x - ab]
= 6a²b²- (12-10)abxy - 20x²y²
= 6a²b²- 2abxy - 20x²y² (Ans.)
D) (x-a)(x-b) = x²- (a+b).x + ab
Proof - (x-a)(x-b) = (x-a)x – (x-a)b
= x² -ax – bx +ab
= x² - (a+b)x + ab (Proven)
There are some example are given below for your better understanding -
Example.1) (x-5)(x-4) = x²- (5+4)x + 5.4
[applying formula (x-a).(x-b) = x² - (a+b)x + ab]
= x² - 9x + 20 (Ans.)
Example.2) (2x-3)(3x-7) = 2.3x²- (3.3+2.7)x + 3.7
[applying formula (x-a).(x-b) = x² - (a+b)x + ab]
= 6x²- (9+14)x +21
= 6x² - 23x + 21 (Ans.)
Example.3) (2x-3a)(3x-4a) = 2.3x²- (3.3+2.4)ax + 3.4a²
[applying formula (x-a).(x-b) = x² - (a+b).x + ab]
= 6x²- (9+8)ax + 12a²
= 6x²- 17ax + 12a² (Ans.)
Example.4) (5ab-2xy)(4ab-5xy)
= 5.4.(ab)²- (2.4+5.5)ab.xy + 2.5.(xy)²
[applying formula (x+a).(x-b) = x² + (a-b).x – ab]
= 20a²b²- (8+25)abxy + 10x²y²
= 20a²b²- 33abxy + 10x²y² (Ans.)