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Answer :
To calculate [tex]\( P(11) \)[/tex] for the polynomial [tex]\( P(x) = x^4 - 23x^2 + 18x + 40 \)[/tex] using synthetic division and the Remainder Theorem, follow these steps:
1. Identify the Coefficients:
The polynomial [tex]\( P(x) = x^4 - 23x^2 + 18x + 40 \)[/tex] can be expressed with the coefficients:
[tex]\([1, 0, -23, 18, 40]\)[/tex].
Note that the coefficient for [tex]\( x^3 \)[/tex] is missing in the polynomial, so we use 0 in its place.
2. Synthetic Division Process:
- Start with the first coefficient, which is 1.
- Multiply this coefficient by the value we're evaluating at, [tex]\( 11 \)[/tex], and add the result to the next coefficient.
- Continue this process across the row of coefficients.
Here’s a step-by-step breakdown:
- Bring down the first coefficient:
[tex]\( 1 \)[/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the next coefficient (which is [tex]\( 0 \)[/tex]):
[tex]\[
1 \times 11 + 0 = 11
\][/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the coefficient [tex]\( -23 \)[/tex]:
[tex]\[
11 \times 11 + (-23) = 121 - 23 = 98
\][/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the coefficient [tex]\( 18 \)[/tex]:
[tex]\[
98 \times 11 + 18 = 1078 + 18 = 1096
\][/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the last coefficient [tex]\( 40 \)[/tex]:
[tex]\[
1096 \times 11 + 40 = 12056 + 40 = 12096
\][/tex]
3. Conclusion:
The remainder from the synthetic division gives us the value of the polynomial at [tex]\( x = 11 \)[/tex]. Therefore, [tex]\( P(11) = 12096 \)[/tex].
Hence, using the Remainder Theorem and synthetic division, we find that the remainder, which is equivalent to [tex]\( P(11) \)[/tex], is 12096.
1. Identify the Coefficients:
The polynomial [tex]\( P(x) = x^4 - 23x^2 + 18x + 40 \)[/tex] can be expressed with the coefficients:
[tex]\([1, 0, -23, 18, 40]\)[/tex].
Note that the coefficient for [tex]\( x^3 \)[/tex] is missing in the polynomial, so we use 0 in its place.
2. Synthetic Division Process:
- Start with the first coefficient, which is 1.
- Multiply this coefficient by the value we're evaluating at, [tex]\( 11 \)[/tex], and add the result to the next coefficient.
- Continue this process across the row of coefficients.
Here’s a step-by-step breakdown:
- Bring down the first coefficient:
[tex]\( 1 \)[/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the next coefficient (which is [tex]\( 0 \)[/tex]):
[tex]\[
1 \times 11 + 0 = 11
\][/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the coefficient [tex]\( -23 \)[/tex]:
[tex]\[
11 \times 11 + (-23) = 121 - 23 = 98
\][/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the coefficient [tex]\( 18 \)[/tex]:
[tex]\[
98 \times 11 + 18 = 1078 + 18 = 1096
\][/tex]
- Multiply by [tex]\( 11 \)[/tex], then add the last coefficient [tex]\( 40 \)[/tex]:
[tex]\[
1096 \times 11 + 40 = 12056 + 40 = 12096
\][/tex]
3. Conclusion:
The remainder from the synthetic division gives us the value of the polynomial at [tex]\( x = 11 \)[/tex]. Therefore, [tex]\( P(11) = 12096 \)[/tex].
Hence, using the Remainder Theorem and synthetic division, we find that the remainder, which is equivalent to [tex]\( P(11) \)[/tex], is 12096.
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