Question

#Quality question need a well explained answer


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.​

Answer 1

Answer:

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.

Step-by-step explanation:

#Queen

Answer 2

$\huge\purple{\mathbb{Answer}}$

• 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 -

$\frac{p(x)}{x + k} = q(x) + \frac{r}{x + k}$

• 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) =

$\frac{ - 2}{5}$

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.