Find the exact zeros (show your work using synthetic division and\ or factoring )F(x)=x^3-7x^2+13x-3.

x^3 - 5x^2 +8x-4=0 Answer is 1 or 2 but having difficulty figuring out how to solve this other than trial and error.

We could use DesCartes rule of signs, to show 3 positive roots, then the rational zeros test to narrow the choices. Finally using synthetic division to get them. However, since we know them and they are rational, let's factor.


Then...

What are the rational routes of 8x^3+2x^2-5x+1=0 ?

By using synthetic division we can write:
8x^3+2x^2-5x+1=(x+1)(8x^2-6x+1)
then by the quadratic equation, factoring, or some other method we get
8x^3+2x^2-5x+1=(x+1)(2x-1)(4x-1)
The roots are: {-1, 1/2 1/4}

Factor 2x^3+x^2+12x+6

2x^3+x^2+12x+6=
2x^3+12x+x^2+6=
(2x^3+12x)+(x^2+6)=
2x(x^2+6)+(x^2+6)=
(2x+1)(x^2+6)

Factorize m^4+m^2+1

Factoring isn't easy, until you get good at it. One useful technique is to introduce terms that add to 0 as follows:




now we can write...

rearranging and grouping leads to

find the zeros fro g(x), g(x) = 4(x + 2)^5 (x-3)^3 + 5(x-3)^4 (x+2)^4

by grouping 
 
becomes 
 

which simplifies to 
 
so there are 3 unique solutions: 
x=-2 (multiplicity 4)
x=3 (multiplicity 3)
x=7/9

Factor 12xy+8x-15y-10

12xy+8x-15y-10= 

(12xy+8x)-(15y+10)= 


4x(3y+2)-5(3y+2)= 


(4x-5)(3y+2) 


:)

Factor:

Factor

Factor quadratic trinomials by grouping

Factoring Polynomials

factor 12ac+21ad+8bc+14bd

Factor the polynomial.