Answer :
To find the value of [tex]\( f(-3) \)[/tex], we use the synthetic division method shown in the table. Here's a step-by-step explanation:
1. Understand the table setup: The table represents a division of the polynomial [tex]\( f(x) \)[/tex] by the binomial [tex]\( x + 3 \)[/tex]. In synthetic division, you use the opposite sign of the number being divided by, so you use -3.
2. Coefficients of the polynomial: The first row of numbers, 2, -5, and 3, represents the coefficients of the polynomial [tex]\( f(x) \)[/tex].
3. Bring down the first coefficient: Start by bringing down the first coefficient directly below the line, which is 2.
4. Multiply and add:
- Multiply the number outside the division box (-3) by the number you just brought down (2), which gives you -6. Write this value under the second coefficient.
- Add the second coefficient (-5) and the product (-6) to get -11. Write this result in the third column below the line.
5. Repeat multiply and add:
- Multiply -3 by -11 to get 33. Write this value under the third coefficient (3).
- Add the third coefficient (3) and 33 to get 36.
6. Identify the remainder: The final number in the bottom row, 36, is the remainder of the division. In synthetic division, this remainder is the value of the polynomial evaluated at the divisor, which is [tex]\( f(-3) \)[/tex].
Therefore, the value of [tex]\( f(-3) \)[/tex] is 36.
1. Understand the table setup: The table represents a division of the polynomial [tex]\( f(x) \)[/tex] by the binomial [tex]\( x + 3 \)[/tex]. In synthetic division, you use the opposite sign of the number being divided by, so you use -3.
2. Coefficients of the polynomial: The first row of numbers, 2, -5, and 3, represents the coefficients of the polynomial [tex]\( f(x) \)[/tex].
3. Bring down the first coefficient: Start by bringing down the first coefficient directly below the line, which is 2.
4. Multiply and add:
- Multiply the number outside the division box (-3) by the number you just brought down (2), which gives you -6. Write this value under the second coefficient.
- Add the second coefficient (-5) and the product (-6) to get -11. Write this result in the third column below the line.
5. Repeat multiply and add:
- Multiply -3 by -11 to get 33. Write this value under the third coefficient (3).
- Add the third coefficient (3) and 33 to get 36.
6. Identify the remainder: The final number in the bottom row, 36, is the remainder of the division. In synthetic division, this remainder is the value of the polynomial evaluated at the divisor, which is [tex]\( f(-3) \)[/tex].
Therefore, the value of [tex]\( f(-3) \)[/tex] is 36.
