Special Products of Polynomials

1.) [5(2a-b)-3][5(2a-b)+3]

2.) [5(2a-b)-3][5(2a-b)-3]

3.) (a-4b+c)(a+4b+c)

4.) (a-4b-c)(a+4b+c)

1.) [5(2a-b)-3][5(2a-b)+3]
= [10a-5b-3][10a-5b+3]
=10a[10a-5b+3]-5b[10a-5b+3]-3[10a-5b+3…
=100a^2-50ab+30a-50ab+25b^2-15b-30a+15…
=100a^2-100ab+25b^2-9
Notice that we can arrange the terms so as to obtain 'difference of two squares':
(a-b)(a+b) = a^2 - b^2
In practice:
[5(2a-b)-3][5(2a-b)+3]
=[(10a-5b)-3][(10a-5b)+3]
Using difference of two squares, we obtain:
=(10a-5b)^2 -(3)^2
[Now, (a-b)^2 = a^2 - 2ab + b^2]
=100a^2-2(10a)(5b)+25b^2 - 9
=100a^2 - 100ab + 25b^2 - 9
2.) [5(2a-b)-3][5(2a-b)-3]
= [ (10a-5b) - 3 ] [ (10a - 5b) - 3 ]
= [ (10a-5b) - 3 ] ^ 2
We use the relation (a-b)^2 = a^2-2ab+b^2
= (10a-5b)^2 - 2(10a -5b)(3) + 3^2
= (100a^2-100ab+25b^2) - (60a-30b) + 9
= 100a^2+25b^2-100ab-60a+39b+9
3.) (a-4b+c)(a+4b+c)
Rearranging this to:
[ (a+c) -4b] [ (a+c) +4b ]
We now apply 'difference of two squares':
= (a+c)^2 - (4b)^2
= (a^2 + 2ac + c^2) - 16b^2
= a^2 - 16b^2 + c^2 - 2ac
4.) (a-4b-c)(a+4b+c)
Rearranging this:
[a - (4b + c)] [a + (4b + c)]
Using difference of two squares:
= a^2 - (4b+c)^2
= a^2 - [ (4b)^2 + 2(4b)(c) + c^2 ]
= a^2 - ( 16b^2 +8bc + c^2 ]
=a^2 - 16b^2 - c^2 - 8bc