High School

When the polynomial \( p(x) \) is evaluated at \( x = 9 \), the result is 6. Which statement MUST be correct?

A) If \( p(x) \) is divided by \( x - 9 \), the remainder is 6.
B) If \( p(x) \) is divided by \( x - 6 \), the remainder is 9.
C) If \( p(x) \) is divided by \( x + 6 \), the remainder is 9.
D) If \( p(x) \) is divided by \( x + 9 \), the remainder is 6.

Answer :

Any polynomial f(x) will have a root "a" if and only if it satisfy the condition :- f(a)=0 i.e. Factor (x-a) will divide f(x) completely leaving remainder 0.

If we get f(b) = k, where b and k are some integer values. Then 'k' is the remainder after dividing f(x) by (x-b).

Similarly, if we get p(9)=6 for any polynomial p(x). Then we must get 6 as remainder after dividing p(x) by factor (x-9).

Hence, option A is correct i.e. If p(x) is divided by x – 9, the remainder is 6.
Answer: Option A) If p(x) is divided by x – 9, the remainder is 6.

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