Answer :
To solve this problem, we need to use the Remainder Theorem. The theorem states that the remainder of the division of a polynomial [tex]\( p(x) \)[/tex] by [tex]\( (x-a) \)[/tex] is equal to [tex]\( p(a) \)[/tex].
Let's apply this theorem to the given information:
We know that when the polynomial [tex]\( p(x) \)[/tex] is evaluated at [tex]\( x = 9 \)[/tex], the result is 6. In mathematical terms, this means [tex]\( p(9) = 6 \)[/tex].
According to the Remainder Theorem, since [tex]\( p(9) = 6 \)[/tex], this implies that if [tex]\( p(x) \)[/tex] is divided by [tex]\( (x-9) \)[/tex], the remainder must be 6. This directly corresponds to option A.
Now let's verify each option:
A. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x-9 \)[/tex], the remainder is 6.
- This statement is correct because [tex]\( p(9) = 6 \)[/tex].
B. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x-6 \)[/tex], the remainder is 9.
- There is no information provided to support this. This statement is incorrect.
C. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x+6 \)[/tex], the remainder is 9.
- Similarly, there is no information about [tex]\( p(-6) \)[/tex]. This statement is incorrect.
D. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x+9 \)[/tex], the remainder is 6.
- Once again, there is no information about [tex]\( p(-9) \)[/tex]. This statement is incorrect.
Therefore, the correct choice is A: If [tex]\( p(x) \)[/tex] is divided by [tex]\( x-9 \)[/tex], the remainder is 6.
Let's apply this theorem to the given information:
We know that when the polynomial [tex]\( p(x) \)[/tex] is evaluated at [tex]\( x = 9 \)[/tex], the result is 6. In mathematical terms, this means [tex]\( p(9) = 6 \)[/tex].
According to the Remainder Theorem, since [tex]\( p(9) = 6 \)[/tex], this implies that if [tex]\( p(x) \)[/tex] is divided by [tex]\( (x-9) \)[/tex], the remainder must be 6. This directly corresponds to option A.
Now let's verify each option:
A. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x-9 \)[/tex], the remainder is 6.
- This statement is correct because [tex]\( p(9) = 6 \)[/tex].
B. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x-6 \)[/tex], the remainder is 9.
- There is no information provided to support this. This statement is incorrect.
C. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x+6 \)[/tex], the remainder is 9.
- Similarly, there is no information about [tex]\( p(-6) \)[/tex]. This statement is incorrect.
D. If [tex]\( p(x) \)[/tex] is divided by [tex]\( x+9 \)[/tex], the remainder is 6.
- Once again, there is no information about [tex]\( p(-9) \)[/tex]. This statement is incorrect.
Therefore, the correct choice is A: If [tex]\( p(x) \)[/tex] is divided by [tex]\( x-9 \)[/tex], the remainder is 6.
