Answer :
To determine the value of [tex]\( f(-3) \)[/tex] when using synthetic division to divide the polynomial [tex]\( f(x) \)[/tex] by [tex]\( x + 3 \)[/tex], we can apply the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by a linear divisor [tex]\( x - c \)[/tex] is equal to [tex]\( f(c) \)[/tex].
In this case, we are dividing by [tex]\( x + 3 \)[/tex], which can be rewritten as [tex]\( x - (-3) \)[/tex]. According to the Remainder Theorem, the value of the polynomial [tex]\( f(x) \)[/tex] at [tex]\( x = -3 \)[/tex] is exactly the remainder of this division.
Here’s a step-by-step breakdown of the process:
1. Set Up for Synthetic Division:
- The divisor is [tex]\( x + 3 \)[/tex], which means we substitute [tex]\( x = -3 \)[/tex] into the polynomial.
2. Perform Synthetic Division:
- Write down the coefficients of the polynomial.
- Use [tex]\(-3\)[/tex] as the number at the top of synthetic division.
- Perform synthetic division by bringing down the first coefficient, multiply by [tex]\(-3\)[/tex], and add to the next coefficient. Repeat this process through all coefficients.
3. Identify the Remainder:
- The result of the synthetic division will provide a quotient and a remainder.
- The remainder is the rightmost value in the synthetic division process. This remainder represents [tex]\( f(-3) \)[/tex].
4. Conclusion:
- After completing the synthetic division process, we find the remainder.
- Therefore, the value of [tex]\( f(-3) \)[/tex] is 36.
Thus, based on this method and analysis, [tex]\( f(-3) \)[/tex] is 36.
In this case, we are dividing by [tex]\( x + 3 \)[/tex], which can be rewritten as [tex]\( x - (-3) \)[/tex]. According to the Remainder Theorem, the value of the polynomial [tex]\( f(x) \)[/tex] at [tex]\( x = -3 \)[/tex] is exactly the remainder of this division.
Here’s a step-by-step breakdown of the process:
1. Set Up for Synthetic Division:
- The divisor is [tex]\( x + 3 \)[/tex], which means we substitute [tex]\( x = -3 \)[/tex] into the polynomial.
2. Perform Synthetic Division:
- Write down the coefficients of the polynomial.
- Use [tex]\(-3\)[/tex] as the number at the top of synthetic division.
- Perform synthetic division by bringing down the first coefficient, multiply by [tex]\(-3\)[/tex], and add to the next coefficient. Repeat this process through all coefficients.
3. Identify the Remainder:
- The result of the synthetic division will provide a quotient and a remainder.
- The remainder is the rightmost value in the synthetic division process. This remainder represents [tex]\( f(-3) \)[/tex].
4. Conclusion:
- After completing the synthetic division process, we find the remainder.
- Therefore, the value of [tex]\( f(-3) \)[/tex] is 36.
Thus, based on this method and analysis, [tex]\( f(-3) \)[/tex] is 36.
