Question

Fill in the blanks:
If 2 is a zero of the polynomial $a{x^2} - 2x$, then the value of ‘a’ is ………………

Answer 1

Hint: According to the Factor theorem:
If 2 is a zero of the polynomial, then it means that if we substitute the variable with 2 in the polynomial , it will satisfy the polynomial to give the answer as 0.that is $a{x^2} - 2x = 0$, with x = 2.

Complete step-by-step answer:
According to the question, we need to find the valueof a for the polynomial $a{x^2} - 2x$, which has 2 as a zero.
Let $p(x) = a{x^2} - 2x$, be the given polynomial.
Now, if 2 is a zero of the polynomial$p(x)$, it means that It satisfies the polynomial $p(x)$ so that $p(x) = 0$when we put x = 2 in $p(x)$.
Zero or root or factor of a polynomial $p(x)$ is or are that value, which when put in the place of the variables, the value of the polynomial $p(x)$ becomes 0. Hence, this is how they become the solution of the polynomial $p(x)$ by satisfying the polynomial$p(x)$.
Now, according to the question, 2 is a zero of the polynomial $p(x) = 0$$ \Rightarrow a{x^2} - 2x = 0$, now, putting $x = 2$ in $a{x^2} - 2x = 0$, the polynomial will become:
$
  p(2) = 0 \\
   \Rightarrow a{(2)^2} - 2(2) = 0 \\
   \Rightarrow 4a - 4 = 0 \\
   \Rightarrow 4a = 4 \\
   \Rightarrow a = \dfrac{4}{4} \\
   \Rightarrow a = 1 \\
$
Hence the value of a for which it is a zero for the polynomial $a{x^2} - 2x$ is 1.
Therefore, the value of ‘a’ is 1.

Note: Here we are using the statement of Factor theorem since the factor is given to us. The converse of Factor theorem is used when we need to find the factor from a given polynomial $p(x)$ . The converse of Factor theorem says, that if there exists a factor ‘a’, such that when we put x = a in the polynomial $p(x)$then the value of the polynomial $p(x)$will become 0, then ‘a’ is a zero or root or factor of polynomial $p(x)$.