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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]\(\square\)[/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]



Answer :

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.