Factor [& remainder] theorem

Question 1

Equation to know when factorising f(x)?

Answer 1

f(x)(ax^2+bx+c)

Question 2

After timesing, what ya do?

Answer 2

Factorise within e.g.
If we did (x-1)(ax^2+bx+c),
ax^3+bx+cx-ax^2-bx-c
Rearrange em and do tha thing

ax^3+(b-a)x^2+(c-b)x-c

u align them with f(x) =

ax^3+(b-a)x^2+(c-b)x-c = 2x^3+x^2-5x+2
rq we found val of c from previous question, c = 2)

1) ax^3 = 2x^3
a = 2

2) -c = 2
c = -2

3) (b-a)x^2 = x^2
divide all by x^2
(b-a) = 1
b-2 = 1
b = 3

Final part:
a = 2, b = 3, c = -2
Put them in an equation:
2x^2+3x-2
Then just factorise it:
(2x-1)(x+2)

Now we have all 3 solutions:
(x-1)(2x-1)(x+2)

(btw, careful using calculator for that lil 2x-1 thing cuz i got x = 1/2)

Question 3

So what about remainder theorem? (following on from previous question btw)

Find remainder when f(x) / (2x-3)

Answer 3

U quite literally just get value of x from (2x-3):
2x-3 = 0
2x = 3
x = 1.5

Sub in 1.5 to f(x):
f(1.5) = 2(1.5)^3+1.5^2-5(1.5)+2
= 7/2

Just a heads up, f(x) was 2x^3 + x^2 - 5x + 2