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Question

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x)=2x3+x22x1,g(x)=x+1
(ii) p(x)=x3+3x2+3x+1,g(x)=x+2
(iii) p(x)=x34x2+x+6,g(x)=x3

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Solution

(i) Apply factor theorem

x+1=0

So x=1

2x3+x22x1

Replace x by 1, we get

2(1)3+(1)22(1)1=2+1+21=0

Reminder is 0 so that x+1 is a factor of 2x3+x22x1

(ii) Apply factor theorem

x+2=0

So x=2

x3+3x2+3x+1

Replace x by 2, we get

(2)3+3(2)2+3(2)+1=8+126+1=1

Reminder is 1 so that x+2 is not a factor of x3+3x2+3x+1

(iii) Apply factor theorem

x3=0

So x=3

x34x2+x+6

Replace x by 3, we get

(3)34(3)2+(3)+6=2736+3+6=0

Reminder is 0 so that x3 is a factor of x34x2+x+6

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