To solve the problem, we assess the provided polynomial [tex]\( p(x) = x^4 + 5x^3 + ax^2 - 3x + 11 \)[/tex]. We are informed that the remainder when [tex]\( p(x) \)[/tex] is divided by [tex]\( x+1 \)[/tex] is 17.
Braulio’s approach:
Using synthetic division to find the value of [tex]\( a \)[/tex]:
1. Rewrite [tex]\( x+1 \)[/tex] as [tex]\( x - (-1) \)[/tex].
2. Use synthetic division by substituting [tex]\( x = -1 \)[/tex] into [tex]\( p(x) \)[/tex].
Zahra’s approach:
Using the remainder theorem to find the value of [tex]\( a \)[/tex]:
1. Substitute [tex]\( x = -1 \)[/tex] into the polynomial [tex]\( p(x) \)[/tex].
2. According to the remainder theorem, [tex]\( p(-1) \)[/tex] should equal the remainder, here known to be 17:
[tex]\[
p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11
\][/tex]
Simplifying this:
[tex]\[
p(-1) = 1 - 5 + a + 3 + 11
\][/tex]
[tex]\[
p(-1) = 1 - 5 + 3 + 11 + a
\][/tex]
[tex]\[
p(-1) = 10 + a
\][/tex]
Since we know that [tex]\( p(-1) = 17 \)[/tex]:
[tex]\[
10 + a = 17
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 17 - 10
\][/tex]
[tex]\[
a = 7
\][/tex]
Both Braulio and Zahra found the value of [tex]\( a \)[/tex] correctly following their respective methods.
Given this, the correct statements would be:
- Braulio correctly found the value of [tex]\( a \)[/tex] because he used synthetic division.
- Zahra correctly found the value of [tex]\( a \)[/tex] because she used the remainder theorem.
Thus, the completed sentences should be:
Braulio correctly found the value of [tex]\( a \)[/tex] because he used synthetic division.
Zahra correctly found the value of [tex]\( a \)[/tex] because she used the remainder theorem.
