Answer :
Sure! To calculate [tex]\( P(16) \)[/tex] using synthetic division and the Remainder Theorem, let's go through the process step-by-step.
Polynomial: [tex]\( P(x) = x^4 - 4x^3 - 23x^2 - 54x + 72 \)[/tex]
Remainder Theorem: This theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder of that division is [tex]\( P(c) \)[/tex].
Given function: [tex]\( P(x) \)[/tex], we want to find [tex]\( P(16) \)[/tex].
### Synthetic Division Process
1. Write down the coefficients:
- For [tex]\( P(x) = x^4 - 4x^3 - 23x^2 - 54x + 72 \)[/tex], the coefficients are [tex]\( 1, -4, -23, -54, 72 \)[/tex].
2. Set up for synthetic division:
- We will perform synthetic division with [tex]\( c = 16 \)[/tex].
3. Begin synthetic division:
- Start with the leading coefficient (1).
- Multiply it by 16 and add it to the next coefficient.
4. Perform each step:
- Step 1: Bring down the 1.
- Step 2: Multiply 1 by 16 (result: 16) and add to -4 (result: 12).
- Step 3: Multiply 12 by 16 (result: 192) and add to -23 (result: 169).
- Step 4: Multiply 169 by 16 (result: 2704) and add to -54 (result: 2650).
- Step 5: Multiply 2650 by 16 (result: 42400) and add to 72 (result: 42472).
5. Find the remainder:
- The final number obtained in the synthetic division process is the remainder, which by the Remainder Theorem, is [tex]\( P(16) \)[/tex].
Result:
Therefore, [tex]\( P(16) = 42472 \)[/tex].
This result indicates that when you substitute [tex]\( x = 16 \)[/tex] into the polynomial [tex]\( P(x) \)[/tex], the value is 42472.
Polynomial: [tex]\( P(x) = x^4 - 4x^3 - 23x^2 - 54x + 72 \)[/tex]
Remainder Theorem: This theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder of that division is [tex]\( P(c) \)[/tex].
Given function: [tex]\( P(x) \)[/tex], we want to find [tex]\( P(16) \)[/tex].
### Synthetic Division Process
1. Write down the coefficients:
- For [tex]\( P(x) = x^4 - 4x^3 - 23x^2 - 54x + 72 \)[/tex], the coefficients are [tex]\( 1, -4, -23, -54, 72 \)[/tex].
2. Set up for synthetic division:
- We will perform synthetic division with [tex]\( c = 16 \)[/tex].
3. Begin synthetic division:
- Start with the leading coefficient (1).
- Multiply it by 16 and add it to the next coefficient.
4. Perform each step:
- Step 1: Bring down the 1.
- Step 2: Multiply 1 by 16 (result: 16) and add to -4 (result: 12).
- Step 3: Multiply 12 by 16 (result: 192) and add to -23 (result: 169).
- Step 4: Multiply 169 by 16 (result: 2704) and add to -54 (result: 2650).
- Step 5: Multiply 2650 by 16 (result: 42400) and add to 72 (result: 42472).
5. Find the remainder:
- The final number obtained in the synthetic division process is the remainder, which by the Remainder Theorem, is [tex]\( P(16) \)[/tex].
Result:
Therefore, [tex]\( P(16) = 42472 \)[/tex].
This result indicates that when you substitute [tex]\( x = 16 \)[/tex] into the polynomial [tex]\( P(x) \)[/tex], the value is 42472.
