Question

If (x-1) is a factor of polynomial f(x) and 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

Hint: For a polynomial, there could be some values of the variable for which the polynomial will be zero. These values are called zeros of a polynomial. Sometimes, they are also referred to as roots of the polynomials, In general, we use them to find the zeros of quadratic equations, to get the solutions for the given equation.

Complete step-by-step answer:
Knowing the above mentioned definition of zeros of a polynomial, we can easily find the answer of the given question.
Also, on equating an expression to 0, we get the zero of the polynomial of which the given expression is a factor.
As mentioned in the question, we have to find the option of which (x-1) is a factor.
Now, on equating the factor with 0, we get the following expression
(x-1)=0
x=1
So, we get that x=1 is a zero or root of the function f(x). Hence, we can say that f(1)=0.
Now, by putting x=1 in every option, we can find out which is the option of which (x-1) is a factor.
\[\begin{align}
  & (a)\ f(1)\cdot g(1)=0 \\
 & (b)\ -f(1)+g(1)=g(1) \\
 & (c)\ f(1)-g(1)=-g(1) \\
 & (d)\ \{f(1)+g(1)\}\cdot g(1)={{\left( g(1) \right)}^{2}} \\
\end{align}\]
Hence, option (a) is correct.

Note:-The students can make an error if they don’t know about the definitions and the meaning of zeros of a polynomial or zeros of a function as without knowing the definition of zeros of a function; one can never get to the correct answer.
Also, option elimination can also be done to find the correct option faster