Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the value of [tex]\(a\)[/tex] in the polynomial [tex]\( p(x) = x^4 + 5x^3 + ax^2 - 3x + 11 \)[/tex] when the remainder of the division by [tex]\((x+1)\)[/tex] is 17, we can use two different methods: synthetic division and the remainder theorem. Let's see how Braulio and Zahra approached this problem.
### Braulio's Method: Synthetic Division
1. Set up synthetic division for the polynomial [tex]\( p(x) \)[/tex] with the divisor [tex]\((x + 1)\)[/tex], which has the root [tex]\(-1\)[/tex].
The coefficients of [tex]\( p(x) \)[/tex] are [tex]\([1, 5, a, -3, 11]\)[/tex].
2. Perform synthetic division using [tex]\(-1\)[/tex]:
[tex]\[ \begin{array}{r|rrrrr} & 1 & 5 & a & -3 & 11 \\ -1 & & & & & \\ \hline & 1 & 4 & a-4 & -7 & 17 \\ \end{array} \][/tex]
Here’s the step-by-step calculation:
[tex]\[ \begin{array}{r|rrrrr} & 1 & 5 & a & -3 & 11 \\ -1 & & -1 & -4 & -(a-4) & -(a-14) \\ \hline & 1 & 4 & a-4 & -7 & a-14 \\ \end{array} \][/tex]
- The first coefficient is 1; it goes directly down.
- Multiply -1 (root) with 1 (first coefficient) and add to 5 (second coefficient): [tex]\(5 - 1 = 4\)[/tex].
- Multiply -1 with 4 (obtained value) and add to [tex]\(a\)[/tex]: [tex]\(a - 4\)[/tex].
- Multiply -1 with [tex]\(a-4\)[/tex] and add to -3: [tex]\(-3 - (a-4) = -3 - a + 4 = 1 - a\)[/tex].
- Multiply -1 with [tex]\(1 - a\)[/tex] and add to 11: [tex]\(11 - (1 - a) = 11 - 1 + a = 10 + a\)[/tex].
The last value is the remainder when divided by [tex]\((x+1)\)[/tex].
We are given that this remainder is 17:
[tex]\[ a - 14 = 17 \implies a = 17 + 14 = 31 \][/tex]
Performing a quick check:
- [tex]\(1 \rightarrow \text{drops down as } 1\)[/tex]
- [tex]\(5 + (-1 \times 1) = 5 - 1 = 4\)[/tex]
- [tex]\(a + (-1 \times 4) = a - 4\)[/tex]
- [tex]\(-3 + (-1 \times (a-4)) = -3 - a + 4 = 1 - a\)[/tex]
- [tex]\(11 + (-1 \times (1 - a)) = 11 - 1 + a = 10 + a\)[/tex]
3. Therefore, Braulio's value for [tex]\(a\)[/tex] is [tex]\(31\)[/tex].
### Zahra's Method: Remainder Theorem
1. According to the remainder theorem, if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder is [tex]\( f(c) \)[/tex].
Here, we need to find [tex]\( p(-1) \)[/tex]:
[tex]\[ p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11 \][/tex]
2. Substitute [tex]\(-1\)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[ \begin{aligned} p(-1) &= 1 + 5(-1) + a(1) - 3(-1) + 11 \\ &= 1 - 5 + a + 3 + 11 \\ &= 1 - 5 + 3 + 11 + a \\ &= 1 - 5 + 14 + a \\ &= 10 + a \end{aligned} \][/tex]
3. We are given [tex]\( p(-1) = 17 \)[/tex], so set up the equation:
[tex]\[ 10 + a = 17 \implies a = 17 - 10 = 7 \][/tex]
### Conclusion:
- Braulio, using synthetic division, correctly found the value of [tex]\(a\)[/tex] to be [tex]\(31\)[/tex].
- Zahra, using the remainder theorem, correctly found the value of [tex]\(a\)[/tex] to be [tex]\(7\)[/tex].
### Braulio's Method: Synthetic Division
1. Set up synthetic division for the polynomial [tex]\( p(x) \)[/tex] with the divisor [tex]\((x + 1)\)[/tex], which has the root [tex]\(-1\)[/tex].
The coefficients of [tex]\( p(x) \)[/tex] are [tex]\([1, 5, a, -3, 11]\)[/tex].
2. Perform synthetic division using [tex]\(-1\)[/tex]:
[tex]\[ \begin{array}{r|rrrrr} & 1 & 5 & a & -3 & 11 \\ -1 & & & & & \\ \hline & 1 & 4 & a-4 & -7 & 17 \\ \end{array} \][/tex]
Here’s the step-by-step calculation:
[tex]\[ \begin{array}{r|rrrrr} & 1 & 5 & a & -3 & 11 \\ -1 & & -1 & -4 & -(a-4) & -(a-14) \\ \hline & 1 & 4 & a-4 & -7 & a-14 \\ \end{array} \][/tex]
- The first coefficient is 1; it goes directly down.
- Multiply -1 (root) with 1 (first coefficient) and add to 5 (second coefficient): [tex]\(5 - 1 = 4\)[/tex].
- Multiply -1 with 4 (obtained value) and add to [tex]\(a\)[/tex]: [tex]\(a - 4\)[/tex].
- Multiply -1 with [tex]\(a-4\)[/tex] and add to -3: [tex]\(-3 - (a-4) = -3 - a + 4 = 1 - a\)[/tex].
- Multiply -1 with [tex]\(1 - a\)[/tex] and add to 11: [tex]\(11 - (1 - a) = 11 - 1 + a = 10 + a\)[/tex].
The last value is the remainder when divided by [tex]\((x+1)\)[/tex].
We are given that this remainder is 17:
[tex]\[ a - 14 = 17 \implies a = 17 + 14 = 31 \][/tex]
Performing a quick check:
- [tex]\(1 \rightarrow \text{drops down as } 1\)[/tex]
- [tex]\(5 + (-1 \times 1) = 5 - 1 = 4\)[/tex]
- [tex]\(a + (-1 \times 4) = a - 4\)[/tex]
- [tex]\(-3 + (-1 \times (a-4)) = -3 - a + 4 = 1 - a\)[/tex]
- [tex]\(11 + (-1 \times (1 - a)) = 11 - 1 + a = 10 + a\)[/tex]
3. Therefore, Braulio's value for [tex]\(a\)[/tex] is [tex]\(31\)[/tex].
### Zahra's Method: Remainder Theorem
1. According to the remainder theorem, if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder is [tex]\( f(c) \)[/tex].
Here, we need to find [tex]\( p(-1) \)[/tex]:
[tex]\[ p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11 \][/tex]
2. Substitute [tex]\(-1\)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[ \begin{aligned} p(-1) &= 1 + 5(-1) + a(1) - 3(-1) + 11 \\ &= 1 - 5 + a + 3 + 11 \\ &= 1 - 5 + 3 + 11 + a \\ &= 1 - 5 + 14 + a \\ &= 10 + a \end{aligned} \][/tex]
3. We are given [tex]\( p(-1) = 17 \)[/tex], so set up the equation:
[tex]\[ 10 + a = 17 \implies a = 17 - 10 = 7 \][/tex]
### Conclusion:
- Braulio, using synthetic division, correctly found the value of [tex]\(a\)[/tex] to be [tex]\(31\)[/tex].
- Zahra, using the remainder theorem, correctly found the value of [tex]\(a\)[/tex] to be [tex]\(7\)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
