Question

What should be subtracted to the polynomial $ {x^2} - 16x + 30 $ , so that $ 15 $ is the zero of the resulting polynomial?
(a) $ 30 $
(b) $ 14 $
(c) $ 15 $
(d) $ 16 $

Answer 1

Hint: To find what should be subtracted from a given polynomial, we first see that $ 15 $ is either zero of polynomial or not by using the remainder theorem concept. If not then subtract the remainder from the given polynomial obtained by using remainder theorem.

Complete step-by-step answer:
To find what should be subtracted to the polynomial $ {x^2} - 16x + 30 $ , so that $ 15 $ is the zero. We use the remainder theorem concept.
Remainder theorem concept states that if substituting any number in given polynomial and result which is obtained after simplification is called remainder.
But, if there is no number left after simplification or we can say that the remainder is zero, then we can say that the number which we substituted in the polynomial is zero of the polynomial.
So, to make a number which is not a zero of a polynomial we should subtract remainder obtained from the polynomial.
Therefore, we first find reminder that we get on substituting $ 15 $ in given polynomial $ {x^2} - 16x + 30 $
 $
  {\left( {15} \right)^2} - 16\left( {15} \right) + 30 \\
   \Rightarrow 225 - 240 + 30 \\
   \Rightarrow 255 - 240 \\
   \Rightarrow 15 \\
  $
Therefore, from above we see that $ 15 $ is remainder of the polynomial when x = $ 15 $ is substituted.
Also, we can say that $ 15 $ is not zero of the given polynomial $ {x^2} - 16x + 30 $
Now, to make $ 15 $ as zero of the given polynomial we should subtract $ 15 $ from the given polynomial $ {x^2} - 16x + 30 $ as we discussed above in concept of remainder theorem.
Hence, from given option we see that correct option is (C)

So, the correct answer is “Option C”.

Note: If we divide a given number by other number and if nothing is left in end or as remainder then number is said to divisible or divisor is known as factor or zero of given number, but if some number left in remainder and divisor will not be the factor or zero of given number. To make divisor zero, we must subtract reminder from given number and hence remainder will be zero and so divisor will become facto or zero of given number.