# Factor Theorem & Synthetic Division...1

Discussion in 'Other Pre-University Math' started by nycmathguy, Sep 11, 2021.

1. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
Section 2.3
Question 56

Can you do 56 as a guide for me to try a few more on my own?

nycmathguy, Sep 11, 2021
2. ### MathLover1

Joined:
Jun 27, 2021
Messages:
2,989
2,884
56.

f(x)=x^3-52x-96
x=-6
=>(x+6) is a factor

a.

use synthetic division: if (x+1)=0=>x=-1

-6| 1 0 -52 -96
| -6 36 96
|___________________
....1 -6 -16 0 =>resulting coefficients: 1,-6,-16,0

the quotient is x^2-6x-16, and the remainder is 0

factor it:
x^2-6x-16
x^2-8x+2x-16
(x^2-8x)+(2x-16)
x(x-8)+2(x-8)
(x + 2) (x - 8)

so,
f(x)=x^3-52x-96
f(x)=(x+6)(x + 2) (x - 8)

real solutions:
x=-6
x=-2
x=8

MathLover1, Sep 11, 2021
nycmathguy likes this.
3. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
You said:

"use synthetic division: if (x+1)=0=>x=-1"

Where did (x + 1) come from?

nycmathguy, Sep 11, 2021
4. ### MathLover1

Joined:
Jun 27, 2021
Messages:
2,989
2,884
if given root =x=-1, then factor is (x-root)=> in your case root =-1 then factor is (x-(-1))=>(x+1)

MathLover1, Sep 12, 2021
nycmathguy likes this.

Joined:
Jun 27, 2021
Messages:
5,386