Is (x + y) a Factor?

[nycmathguy]

College Algebra
Section R.6

How is 31 done?

 

[MathLover1]

(x^4+3x^3y-3x^2y^2-xy^3+4y^4)/(x+y)

if (x + y), then (x + y)=0 -> x =- y

When a Polynomial f(x), is divided by (x + y), the remainder will be f(-y).

(x + y) will be a factor if and only if f(-y)=0
(-y)^4+3(-y)^3y-3(-y)^2y^2-(-y)y^3+4y^4=0
y^4-3y^4-3y^4+y^4+4y^4=0
6y^4-6y^4=0
0=0 -> true

so, (x + y) is a factor
check:
(x^4+3x^3y-3x^2y^2-xy^3+4y^4)/(x+y)=(x + y) (x^3 + 2 x^2 y - 5 x y^2 + 4y^3)

 

[nycmathguy]

(x^4+3x^3y-3x^2y^2-xy^3+4y^4)/(x+y)

if (x + y), then (x + y)=0 -> x =- y

When a Polynomial f(x), is divided by (x + y), the remainder will be f(-y).

(x + y) will be a factor if and only if f(-y)=0
(-y)^4+3(-y)^3y-3(-y)^2y^2-(-y)y^3+4y^4=0
y^4-3y^4-3y^4+y^4+4y^4=0
6y^4-6y^4=0
0=0 -> true

so, (x + y) is a factor
check:
(x^4+3x^3y-3x^2y^2-xy^3+4y^4)/(x+y)=(x + y) (x^3 + 2 x^2 y - 5 x y^2 + 4y^3)

Wow! All I needed to do was substitute -y for y in the polynomial. Cool. Check out the link below.

Divide By (x - h)