Answer :
Sure! Let's solve this problem step-by-step.
We have the polynomial:
[tex]\[ P(x) = 6x^7 - 39x^6 + 22x^5 - 291x^4 - 60x^3 - 69x^2 + 14x - 138 \][/tex]
We need to calculate [tex]\( P(7) \)[/tex] in two ways: using synthetic division and by direct substitution.
### (a) Synthetic Division
Synthetic division is a simplified way to divide a polynomial by a linear factor of the form [tex]\( x - c \)[/tex]. In this case, we want to evaluate [tex]\( P(7) \)[/tex], which means our [tex]\( c \)[/tex] will be [tex]\( 7 \)[/tex].
1. Write down the coefficients of the polynomial:
[tex]\[ 6, -39, 22, -291, -60, -69, 14, -138 \][/tex]
2. Start with the leading coefficient, which is [tex]\( 6 \)[/tex].
3. Carry down the leading coefficient [tex]\( 6 \)[/tex].
Result: [tex]\( 6 \)[/tex]
4. Multiply the last result by [tex]\( 7\)[/tex], then add the next coefficient [tex]\( -39 \)[/tex].
Calculation: [tex]\( 6 \times 7 + (-39) = 42 - 39 = 3 \)[/tex]
5. Repeat the multiply and add process:
- [tex]\( 3 \times 7 + 22 = 21 + 22 = 43 \)[/tex]
- [tex]\( 43 \times 7 + (-291) = 301 - 291 = 10 \)[/tex]
- [tex]\( 10 \times 7 + (-60) = 70 - 60 = 10 \)[/tex]
- [tex]\( 10 \times 7 + (-69) = 70 - 69 = 1 \)[/tex]
- [tex]\( 1 \times 7 + 14 = 7 + 14 = 21 \)[/tex]
- [tex]\( 21 \times 7 + (-138) = 147 - 138 = 9 \)[/tex]
6. The last value obtained is the remainder, which gives [tex]\( P(7) \)[/tex].
So, [tex]\( P(7) = 9 \)[/tex].
### (b) Direct Substitution
We can directly substitute [tex]\( x = 7 \)[/tex] into the polynomial and evaluate:
[tex]\[ P(7) = 6(7^7) - 39(7^6) + 22(7^5) - 291(7^4) - 60(7^3) - 69(7^2) + 14(7) - 138 \][/tex]
Let's calculate each term:
1. [tex]\( 6 \times 7^7 = 6 \times 823543 = 4941258 \)[/tex]
2. [tex]\( 39 \times 7^6 = 39 \times 117649 = 4588321 \)[/tex]
3. [tex]\( 22 \times 7^5 = 22 \times 16807 = 369754 \)[/tex]
4. [tex]\( 291 \times 7^4 = 291 \times 2401 = 698691 \)[/tex]
5. [tex]\( 60 \times 7^3 = 60 \times 343 = 20580 \)[/tex]
6. [tex]\( 69 \times 7^2 = 69 \times 49 = 3381 \)[/tex]
7. [tex]\( 14 \times 7 = 98 \)[/tex]
8. Trailing constant term: [tex]\(-138\)[/tex]
Now summing all terms:
[tex]\[
P(7) = 4941258 - 4588321 + 369754 - 698691 - 20580 - 3381 + 98 - 138 = 9
\][/tex]
Therefore, by direct substitution, we also find [tex]\( P(7) = 9 \)[/tex].
So, both methods confirm that [tex]\( P(7) = 9 \)[/tex].
We have the polynomial:
[tex]\[ P(x) = 6x^7 - 39x^6 + 22x^5 - 291x^4 - 60x^3 - 69x^2 + 14x - 138 \][/tex]
We need to calculate [tex]\( P(7) \)[/tex] in two ways: using synthetic division and by direct substitution.
### (a) Synthetic Division
Synthetic division is a simplified way to divide a polynomial by a linear factor of the form [tex]\( x - c \)[/tex]. In this case, we want to evaluate [tex]\( P(7) \)[/tex], which means our [tex]\( c \)[/tex] will be [tex]\( 7 \)[/tex].
1. Write down the coefficients of the polynomial:
[tex]\[ 6, -39, 22, -291, -60, -69, 14, -138 \][/tex]
2. Start with the leading coefficient, which is [tex]\( 6 \)[/tex].
3. Carry down the leading coefficient [tex]\( 6 \)[/tex].
Result: [tex]\( 6 \)[/tex]
4. Multiply the last result by [tex]\( 7\)[/tex], then add the next coefficient [tex]\( -39 \)[/tex].
Calculation: [tex]\( 6 \times 7 + (-39) = 42 - 39 = 3 \)[/tex]
5. Repeat the multiply and add process:
- [tex]\( 3 \times 7 + 22 = 21 + 22 = 43 \)[/tex]
- [tex]\( 43 \times 7 + (-291) = 301 - 291 = 10 \)[/tex]
- [tex]\( 10 \times 7 + (-60) = 70 - 60 = 10 \)[/tex]
- [tex]\( 10 \times 7 + (-69) = 70 - 69 = 1 \)[/tex]
- [tex]\( 1 \times 7 + 14 = 7 + 14 = 21 \)[/tex]
- [tex]\( 21 \times 7 + (-138) = 147 - 138 = 9 \)[/tex]
6. The last value obtained is the remainder, which gives [tex]\( P(7) \)[/tex].
So, [tex]\( P(7) = 9 \)[/tex].
### (b) Direct Substitution
We can directly substitute [tex]\( x = 7 \)[/tex] into the polynomial and evaluate:
[tex]\[ P(7) = 6(7^7) - 39(7^6) + 22(7^5) - 291(7^4) - 60(7^3) - 69(7^2) + 14(7) - 138 \][/tex]
Let's calculate each term:
1. [tex]\( 6 \times 7^7 = 6 \times 823543 = 4941258 \)[/tex]
2. [tex]\( 39 \times 7^6 = 39 \times 117649 = 4588321 \)[/tex]
3. [tex]\( 22 \times 7^5 = 22 \times 16807 = 369754 \)[/tex]
4. [tex]\( 291 \times 7^4 = 291 \times 2401 = 698691 \)[/tex]
5. [tex]\( 60 \times 7^3 = 60 \times 343 = 20580 \)[/tex]
6. [tex]\( 69 \times 7^2 = 69 \times 49 = 3381 \)[/tex]
7. [tex]\( 14 \times 7 = 98 \)[/tex]
8. Trailing constant term: [tex]\(-138\)[/tex]
Now summing all terms:
[tex]\[
P(7) = 4941258 - 4588321 + 369754 - 698691 - 20580 - 3381 + 98 - 138 = 9
\][/tex]
Therefore, by direct substitution, we also find [tex]\( P(7) = 9 \)[/tex].
So, both methods confirm that [tex]\( P(7) = 9 \)[/tex].
