Find answers to your most challenging questions with the help of IDNLearn.com's experts. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.
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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.
