Select the correct answer from each drop-down menu.
Consider this polynomial, where [tex]\( a \)[/tex] is an unknown real number.
[tex]\[ p(x) = x^4 + 5x^3 + ax^2 - 3x + 11 \][/tex]
The remainder of the quotient of [tex]\( p(x) \)[/tex] and [tex]\( (x + 1) \)[/tex] is 17. Braulio uses synthetic division to find the value of [tex]\( a \)[/tex], and Zahra uses the remainder theorem to find the value of [tex]\( a \)[/tex]. Their work is shown.
Braulio:
[tex]\[
\begin{array}{cccccc}
\text{} & 1 & 5 & a & -3 & 11 \\
1 & 1 & 6 & a+6 & a+3 \\
\hline
1 & 6 & a+6 & a+3 & a+14
\end{array}
\][/tex]
[tex]\[ a + 14 = 17 \][/tex]
[tex]\[ a = 3 \][/tex]
Zahra:
[tex]\[ p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11 \][/tex]
[tex]\[ = 1 - 5 + a + 3 + 11 \][/tex]
[tex]\[ = a + 10 \][/tex]
[tex]\[ p(-1) = 17 \][/tex]
[tex]\[ a + 10 = 17 \][/tex]
[tex]\[ a = 7 \][/tex]
Braulio [tex]\( \square \)[/tex] found the value of [tex]\( a \)[/tex] because he [tex]\( \square \)[/tex].
Zahra [tex]\( \square \)[/tex] found the value of [tex]\( a \)[/tex] because she [tex]\( \square \)[/tex].
