For a polynomial p(x), the value of p(3) is −2. Which of the following must be true about p(x)?

A) x−5 is a factor of p(x).
B) x−2 is a factor of p(x).
C) x+2 is a factor of p(x).
D) The remainder when p(x) is divided by x−3 is −2.

D.) The remainder when p(x) is divided by x−3 is −2.

Explanation:

If the polynomial p(x) is divided by a polynomial of the form x+k (which accounts for all of the possible answer choices in this question), the result can be written as

p(x) /x+k  = q(x) +  r /x+k

where q(x) is a polynomial and r is the remainder. Since x+k is a degree-1 polynomial (meaning it only includes x1 and no higher exponents), the remainder is a real number.

Therefore, p(x) can be rewritten as p(x)=(x+k)q(x)+r, where r is a real number.

The question states that p(3)=−2, so it must be true that

−2=p(3)=(3+k)q(3)+r

Now we can plug in all the possible answers. If the answer is A, B, or C, r will be 0, while if the answer is D, r will be −2.

A. −2=p(3)=(3+(−5))q(3)+0

−2=(3−5)q(3)

−2=(−2)q(3)

This could be true, but only if q(3) = 1

B. −2=p(3)=(3+(−2))q(3)+0

−2=(3−2)q(3)

−2=(−1)q(3)

This could be true, but only if q(3) = 2

C. −2=p(3)=(3+2)q(3)+0

−2=(5)q(3)

This could be true, but only if q(3) =  −2 /5

D. −2=p(3)=(3+(−3))q(3)+(−2)

−2=(3−3)q(3)+(−2)

−2=(0)q(3)+(−2)

This will always be true no matter what q(3) is.

Of the answer choices, the only one that must be true about p(x) is D, that the remainder when p(x) is divided by x−3 is -2.

If the polynomial p(x) is divided by a polynomial of the form x+k (which accounts for all of the possible answer choices in this question), the result can be written as

p(x)

x+k

=q(x)+

r

x+k

where q(x) is a polynomial and r is the remainder. Since x+k is a degree-1 polynomial (meaning it only includes x1 and no higher exponents), the remainder is a real number.

Therefore, p(x) can be rewritten as p(x)=(x+k)q(x)+r, where r is a real number.

The question states that p(3)=−2, so it must be true that

−2=p(3)=(3+k)q(3)+r

Now we can plug in all the possible answers. If the answer is A, B, or C, r will be 0, while if the answer is D, r will be −2.

A. −2=p(3)=(3+(−5))q(3)+0

−2=(3−5)q(3)

−2=(−2)q(3)

This could be true, but only if q(3)=1

B. −2=p(3)=(3+(−2))q(3)+0

−2=(3−2)q(3)

−2=(−1)q(3)

This could be true, but only if q(3)=2

C. −2=p(3)=(3+2)q(3)+0

−2=(5)q(3)

This could be true, but only if q(3)=

−2

5

D. −2=p(3)=(3+(−3))q(3)+(−2)

−2=(3−3)q(3)+(−2)

−2=(0)q(3)+(−2)

This will always be true no matter what q(3) is.

Of the answer choices, the only one that must be true about p(x) is D, that the remainder when p(x) is divided by x−3 is -2.

The final answer is D.

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