Question

How do you factor `x^3+7x^2+14x+8`?

Answer 1

`color(magenta)(=(x+1)(x+4)(x+2)`

Explanation:

`x^3+7x^2+14x+8`

`=x^3-x+7x^2+15x+8` [Adding and subtracting `x`]

`=x(x^2-1)+7x^2+15x+8`

Identity:
`color(red)(a^2-b^2=(a+b)(a-b)` &
`color(red)((x+a)(x+b)=x^2+(a+b)x+ab`

`= x(x+1)(x-1)+[7x(x+1)+8(x+1)`

`= x(x+1)(x-1)+(x+1)(7x+8)`

`= (x+1)[x(x-1)(7x+8)]`

`= (x+1)[x^2-x+7x+8]`

`= (x+1)[x^2+6x+8]`

Identity:
`color(red)((x+a)(x+b)=x^2+(a+b)x+ab`

`color(magenta)(=(x+1)(x+4)(x+2)`

~Hope this helps! :)

Answer 2

(x + 1)(x + 2)(x + 4)

Explanation:

`f(x) = x^3 + 7x^2 + 14x + 8`
First, we note that f(-1) = - 1 + 7 - 14 + 8 = 0.
So, one factor is (x - 1).
After division, or guest -->
`f(x) = (x + 1)(x^2 + 6x + 8)`
Find 2 numbers knowing the sum (6) and the product (8). They are 2 and 4. Finally,
f(x) = (x + 1)(x + 2)(x + 4)