# 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)#?

##### 1 Answer

Evaluate

If the result is

#### 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