Question

select the correct answer from each drop-down menu.

consider this polynomial, where (a) is an unknown real number.
(p(x) = x^4 + 5x^3 + ax^2 - 3x + 11)

the remainder of the quotient of (p(x)) and ((x + 1)) is 17.

braulio uses synthetic division to find the value of (a), and zahra uses the remainder theorem to find the value of (a). their work is shown.

braulio

1 | 1 5 a -3 11
| 1 6 a+6 a+3
-------------------
1 6 a+6 a+3 a+14

(a + 14 = 17)
(a = 3)

zahra

(p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11)
(= -1 + 5 - a + 3 + 11)
(= a + 18)

(p(-1) = 17)
(a + 18 = 17)
(a = -1)

braulio choose an answer found the value of (a) because he choose an answer.
zahra choose an answer found the value of (a) because she choose an answer.

Explanation:

Analyze Braulio's synthetic division

$$
LATEXBLOCK0
$$

Analyze Zahra's remainder theorem calculation

$$
LATEXBLOCK1
$$

Determine the correct drop-down selections

$$
LATEXBLOCK2
$$

Answer:

Braulio <blank>incorrectly</blank> found the value of a because he <blank>used the wrong divisor in his synthetic division setup</blank>.
Zahra <blank>incorrectly</blank> found the value of a because she <blank>made sign errors when evaluating the powers of -1</blank>.