Question

How do you use the remainder theorem and the factor theorem to determine whether (c+5) is a factor of `(c^4+7c^3+6c^2-18c+10)`?

Answer 1

Evaluate `(c^4+7c^3+6c^2-18c+10)` with `c=-5`
If the result is `0` then `(c+5)` is a factor of the given polynomial

Explanation:

The Factor Theorem say:

`(x-a)` is a factor of `f(x)` if and only if `f(a)=0`

The Remainder Theorem says

`f(a)` is the remainder of `f(x)/(x-a)`

For the given expressions this means

`(c+5)` is a factor of `c^4+7c^3+6c^2-18c+10`

if and only if `(color(red)(1)c^4+color(red)(7)c^3+color(red)(6)c^2color(red)(-18)c+color(red)(10))/(c-color(blue)((-5))) = color(green)(0)`

Applying synthetic division:

`{: (," | ",color(red)(1),color(white)("X")color(red)(7),color(white)("XX")color(red)(6),color(red)(-18),color(white)("X")color(red)(10)), (color(blue)(-5)," | ",,-5,-10,color(white)("X") 20,-10), ("---","---","---","-----","------","------","-------"), (,,1,color(white)("X")2,color(white)("X")-4,color(white)("XX")2,color(white)("XX")color(green)(0)) :}`

Giving a remainder of `color(green)(0)`
`rArr (c-(-5)) = (c+5)` is a factor of `c^4+7c^3+6c^2-18c+10`