Question

f(x)=x³+3x²+3x+1,g(x)=x+1,Use the Factor Theorem to determine whether g(x) is factor of f(x) in each cases.

Answer 1

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Factor Theorem :

Let p(x) be a polynomial of

degree one or more than 1

and a is a real number . Then,

i ) x - a , will be a factor of p(x),

if p(a) = 0 ,

Conversely ,

ii ) If x - a is a factor of p(x) , then

p(a) = 0.

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Here ,

F(x) = x³ + 3x² + 3x + 1 ,

g(x) = x + 1 ,

g(x) = 0

=> x + 1 = 0

=> x = -1 ,

Therefore ,

zero of g(x) is -1.

Now ,

F(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

= -1 + 3 - 3 + 1

= -4 + 4

= 0

Therefore ,

By Factor theorem ,

g(x) is a factor of F(x).

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Answer 2

Answer:

$f(x) = x3 + 3x2 + 3x + 1g(x) = 3x - 2$