Question

If (x-1) is a factor of polynomial f(x) but not of g(x), then it must be a factor of
A. f (x) g(x)
B. -f(x)+g(x)
C. f(x)-g(x)
D. {f(x)+g(x)}g(x)

Answer 1

•x-1 is a factor of f(x)g(x)

•As (X-1) is a factor of polynomial f(x), it means it can be divided by (x-1) and it leaves no remainder.

•Now in case of f(x)g(x), it will again get divided and will leave no remainder as X-1 is a factor of f(x).

let say f(X)/x-1 = p(x)

now, f(X)g(X)/x-1= [f(X)/x-1]×g(X)

=p(x)×g(x)

Answer 2

If (x-1) is a factor of polynomial f(x) but not of g(x), then it must be a factor of f (x) g(x)

Step-by-step explanation:

If (x-1) is a factor of polynomial f(x) but not of g(x), then it must be a factor of

f (x) g(x)

f (x) g(x)  will have factors of f(x) & g(x)  so all factors of f(x) will be factors of f(x)g(x) too

Examples to explain:

Let say

f(x) = x² - 1  then x - 1 is factor  of f(x)

g(x) = x²  + 1  then  x - 1 is not a factor  of g(x)

f(x) * g(x) = (x² - 1)(x² + 1)

= (x + 1)(x - 1)(x² + 1)

= (x - 1)(x + 1)(x² + 1)

Hence x - 1 is factor of f(x)g(x)

-f(x) + g(x)

= -x² + 1 + x² + 1

= 2

hence (x -1 ) is not a factor

f(x) - g(x)

= x² - 1 - ( x² + 1)

= - 2

hence (x -1 ) is not a factor

{f(x)+g(x)}g(x)

= (x² - 1  +  x²  + 1)(x² + 1)

= 2x²(x² + 1)

x -1 is not a factor

Hence

If (x-1) is a factor of polynomial f(x) but not of g(x), then it must be a factor of f (x) g(x)

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