Question

select the correct answers from each dropdown.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 17braulio 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$\

$$\begin{array}{r}1 \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\\\\\hline1 \\quad 5 \\quad a \\quad -3 \\quad 11 \\\\\\downarrow \\quad \\quad 1 \\quad 6 \\quad a+6 \\quad a+3 \\\\\\hline1 \\quad 6 \\quad a+6 \\quad a+3 \\quad a+14\\end{array}$$

$$a + 14 = 17$$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 $\boldsymbol{\
abla}$ found the value of a because he $\boldsymbol{\
abla}$zahra $\boldsymbol{\
abla}$ found the value of a because she $\boldsymbol{\
abla}$reset next

Explanation:

First, recall the Remainder Theorem: when dividing a polynomial $p(x)$ by $(x - k)$, the remainder is $p(k)$. Here, we divide by $(x + 1) = (x - (-1))$, so $k=-1$, and the remainder is $p(-1)=17$.

For Braulio's synthetic division: He used $1$ as the root value instead of $-1$ (since we divide by $x+1$, the root is $-1$). This mistake led him to set $a+14=17$ instead of the correct remainder expression.

For Zahra's work: She correctly applied the Remainder Theorem by calculating $p(-1)$, substituted the values properly, solved $a+18=17$, and found $a=-1$, which is mathematically consistent with the remainder condition.

Answer:

Braulio $\boldsymbol{did\ not}$ found the value of $a$ because he $\boldsymbol{used\ the\ wrong\ root\ (1\ instead\ of\ -1)}$
Zahra $\boldsymbol{did}$ found the value of $a$ because she $\boldsymbol{correctly\ applied\ the\ Remainder\ Theorem}$