Find the value of f(3) for the polynomial
𝑓
(
𝑥
)
=
2
𝑥
4
−
5
𝑥
3
−
4
𝑥
+
8
using synthetic substitution

Question

Find the value of f(3) for the polynomial 
𝑓
(
𝑥
)
=
2
𝑥
4
−
5
𝑥
3
−
4
𝑥
+
8
 using synthetic substitution

BlackTom AI · Accepted Answer

We are given the polynomial f(x) = 2x^4 − 5x^3 − 4x + 8 and asked to find f(3) using synthetic substitution.

Step 1: List the coefficients of f(x) in descending powers of x. Since there is no x^2 term, the coefficient sequence is: 2, −5, 0, −4, 8.

Step 2: Set up synthetic substitution with the value 3. Bring down the leading coefficient to start the bottom row: 2.

Step 3: Multiply this brought-down value by 3 and add to the next coefficient:
- 2 × 3 = 6; add to −5 to get 1.

Step 4: Repeat the multiplication and addition process for the remaining coefficients:
- 1 × 3 = 3; add to 0 to get 3.
- 3 × 3 = 9; add to −4 to get 5.
- 5 × 3 = 15; add to 8 to get 23.

Step 5: Interpret the bottom row: the first number (2) is the new leading coefficient after synthetic division, and the final number (23) is the value of the polynomial evaluated at x = 3. The intermediate values correspond to the coefficients of the depressed polynomial of one degree lower, but the key takeaway is that the final remainder equals f(3) in this setup.

Step-by-step recap: the synthetic substitution process with x = 3 proceeds by successive multiplications and additions across the coefficient list [2, −5, 0, −4, 8], producing a final result of 23 as the evaluated value.

Find the value of f(3) for the polynomial 𝑓 ( 𝑥 ) = 2 𝑥 4 − 5 𝑥 3 − 4 𝑥 + 8 using synthetic substitutionNumerical

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