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Let's walk through the steps to solve the problem of finding the correct value of [tex]\(a\)[/tex] in the polynomial [tex]\( p(x) = x^4 + 5x^3 + ax^2 - 3x + 11 \)[/tex], given that the remainder when [tex]\( p(x) \)[/tex] is divided by [tex]\( (x+1) \)[/tex] is 17.
### Braulio's Solution Using Synthetic Division
1. Setup for synthetic division: We are dividing by [tex]\( x + 1 \)[/tex], which corresponds to evaluating at [tex]\( x = -1 \)[/tex].
2. Coefficients of the polynomial: [tex]\( 1, 5, a, -3, 11 \)[/tex]
3. Perform synthetic division:
[tex]\[ \begin{array}{r|rrr} -1 & 1 & 5 & a & -3 & 11 \\ & & -1 & -4 & -a-3 & -a \end{array} \end{array} \][/tex]
- Bring down the 1.
- Multiply by -1 and add to the next coefficient: [tex]\( 5 + (-1) = 4 \)[/tex]
- Multiply by -1 and add to the next coefficient: [tex]\( a + 4 = -a-3 \)[/tex]
- Multiply by -1 and add to the next coefficient: [tex]\(-3 + (-(-a-3)) = -3 + a + 3 = a\)[/tex]
- Multiply by -1 and add to the next coefficient: [tex]\( 11 + (-a) = 11 - a \)[/tex]
4. Final value of the remainder: The last value, [tex]\( a + 14 \)[/tex], is set equal to 17.
5. Set up the equation:
[tex]\[ a + 14 = 17 \][/tex]
6. Solve for [tex]\(a\)[/tex]:
[tex]\[ a = 3 \][/tex]
### Zahra's Solution Using the Remainder Theorem
1. Remainder theorem: Evaluate the polynomial at [tex]\( x = -1 \)[/tex].
2. Setup the polynomial evaluation:
[tex]\[ p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11 \][/tex]
3. Calculate each term:
[tex]\[ (-1)^4 = 1 \][/tex]
[tex]\[ 5(-1)^3 = -5 \][/tex]
[tex]\[ a(-1)^2 = a \][/tex]
[tex]\[ -3(-1) = 3 \][/tex]
[tex]\[ 11 = 11 \][/tex]
4. Sum up the terms:
[tex]\[ p(-1) = 1 - 5 + a + 3 + 11 = a + 10 \][/tex]
5. Set up the equation:
[tex]\[ a + 10 = 17 \][/tex]
6. Solve for [tex]\(a\)[/tex]:
[tex]\[ a = 7 \][/tex]
### Conclusion:
- Braulio found the correct value of [tex]\(a\)[/tex] because he obtained [tex]\( a = 3 \)[/tex], which matches the given answer.
- Zahra did not find the correct value of [tex]\(a\)[/tex] because she obtained [tex]\( a = 7 \)[/tex], which is incorrect.
Therefore, we can say:
- Braulio correctly found the value of [tex]\(a\)[/tex] because he used synthetic division accurately.
- Zahra incorrectly found the value of [tex]\(a\)[/tex] because she made an arithmetic error in her polynomial evaluation.
### Braulio's Solution Using Synthetic Division
1. Setup for synthetic division: We are dividing by [tex]\( x + 1 \)[/tex], which corresponds to evaluating at [tex]\( x = -1 \)[/tex].
2. Coefficients of the polynomial: [tex]\( 1, 5, a, -3, 11 \)[/tex]
3. Perform synthetic division:
[tex]\[ \begin{array}{r|rrr} -1 & 1 & 5 & a & -3 & 11 \\ & & -1 & -4 & -a-3 & -a \end{array} \end{array} \][/tex]
- Bring down the 1.
- Multiply by -1 and add to the next coefficient: [tex]\( 5 + (-1) = 4 \)[/tex]
- Multiply by -1 and add to the next coefficient: [tex]\( a + 4 = -a-3 \)[/tex]
- Multiply by -1 and add to the next coefficient: [tex]\(-3 + (-(-a-3)) = -3 + a + 3 = a\)[/tex]
- Multiply by -1 and add to the next coefficient: [tex]\( 11 + (-a) = 11 - a \)[/tex]
4. Final value of the remainder: The last value, [tex]\( a + 14 \)[/tex], is set equal to 17.
5. Set up the equation:
[tex]\[ a + 14 = 17 \][/tex]
6. Solve for [tex]\(a\)[/tex]:
[tex]\[ a = 3 \][/tex]
### Zahra's Solution Using the Remainder Theorem
1. Remainder theorem: Evaluate the polynomial at [tex]\( x = -1 \)[/tex].
2. Setup the polynomial evaluation:
[tex]\[ p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11 \][/tex]
3. Calculate each term:
[tex]\[ (-1)^4 = 1 \][/tex]
[tex]\[ 5(-1)^3 = -5 \][/tex]
[tex]\[ a(-1)^2 = a \][/tex]
[tex]\[ -3(-1) = 3 \][/tex]
[tex]\[ 11 = 11 \][/tex]
4. Sum up the terms:
[tex]\[ p(-1) = 1 - 5 + a + 3 + 11 = a + 10 \][/tex]
5. Set up the equation:
[tex]\[ a + 10 = 17 \][/tex]
6. Solve for [tex]\(a\)[/tex]:
[tex]\[ a = 7 \][/tex]
### Conclusion:
- Braulio found the correct value of [tex]\(a\)[/tex] because he obtained [tex]\( a = 3 \)[/tex], which matches the given answer.
- Zahra did not find the correct value of [tex]\(a\)[/tex] because she obtained [tex]\( a = 7 \)[/tex], which is incorrect.
Therefore, we can say:
- Braulio correctly found the value of [tex]\(a\)[/tex] because he used synthetic division accurately.
- Zahra incorrectly found the value of [tex]\(a\)[/tex] because she made an arithmetic error in her polynomial evaluation.
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