b). Now that we have a >=1 we multiply a with c
a*c = 4*49 = 196
Now we need ax^2+ nx + dx + c so we need to find two numbers n,d that will satisfy these conditions
Condition 1: n*d = a*c
Condition 2: n+d = b
so we need two numbers n,d that add to 28 and multiply to 196 . Lets try 14,14
(14)(14) = 196 check
14+14 = 28 = b check
4x^2+14x+14x+49 = 2x(2x+7)+7(2x+7)=(2x+7)(2x+7)
so 4x^2+28x+196 = (2x+7)(2x+7).
3).
Rule: x^3-y^3 = (x-y)(x^2+xy+y^2) for a and c
Rule: x^2-y^2 = (x-y)(x+y) for b).
a). 125x^3-8 = (5x)^3-2^3 = (5x-2)(25x^2+10x+4)
b). Factor out x: (x^2-25)(x) = (x+5)(x-5)(x)
c). x^3 - 125 = x^3-5^3 = (x-5)(x^2+5x+25)
d). I don't quite know how to solve for that, look at
http://www.wolframalpha.com/input/?i=+x^… for help
e). Factor x^3 from x^5 + 18x^4 + 81x^3 we get ----> (x^3)(x^2+18x+81)
Find a,b,c in the equation ax^2+bx+c.
x^2+ 18x + 81 ----> a = 1, b = 18, c = 81 since a = 1 we need to find two number n,d that satisfies these conditions
Condition 1: n+d = b(if there is a minus sign in front of b then include it!)
Condition 2: n*d = c(if there is a minus sign in front of c then include it!)
so we need two number that needs to multiply to 81 and add to 18. Lets try 9=n,9=d
Lets check if it satisfies the conditions
9+9 = 18 = b check
9*9 = 81 = c check
now we got the numbers n and d we start step 2
Step 2: ax^2+bx+c = ax^2+nx+dx+c
x^2+18x+81 = x^2+9x+9x+81
Step 3: we factor out things
x(x+9)+9(x+9) = (x+9)(x+9).
So x^5 + 18x^4 + 81x^3 = (x^3)(x^2+18x+81) = x^3(x+9)(x+9)
I am sorry for length and i hope this a good explanation!
Hope it helps Enjoy!