To find the value of [tex]\( A \)[/tex] such that [tex]\( f(2) = 13 \)[/tex] for the polynomial function [tex]\( f(x) = 2x^3 + Ax^2 + 4x - 3 \)[/tex], follow these steps:
1. Start with the given function:
[tex]\[ f(x) = 2x^3 + Ax^2 + 4x - 3 \][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the function since we know [tex]\( f(2) = 13 \)[/tex]:
[tex]\[ f(2) = 2(2)^3 + A(2)^2 + 4(2) - 3 \][/tex]
3. Compute the values inside the function:
[tex]\[ 2(2)^3 = 2 \cdot 8 = 16 \][/tex]
[tex]\[ A(2)^2 = A \cdot 4 = 4A \][/tex]
[tex]\[ 4(2) = 8 \][/tex]
[tex]\[ -3 \text{ (constant term)} \][/tex]
4. Combine these values into the function:
[tex]\[ f(2) = 16 + 4A + 8 - 3 \][/tex]
5. Simplify the above expression:
[tex]\[ f(2) = 21 + 4A \][/tex]
6. Since we know [tex]\( f(2) = 13 \)[/tex], equate this to the simplified expression:
[tex]\[ 21 + 4A = 13 \][/tex]
7. Solve for [tex]\( A \)[/tex] by isolating [tex]\( A \)[/tex]:
[tex]\[ 4A = 13 - 21 \][/tex]
[tex]\[ 4A = -8 \][/tex]
[tex]\[ A = \frac{-8}{4} \][/tex]
[tex]\[ A = -2 \][/tex]
Therefore, the value of [tex]\( A \)[/tex] is:
[tex]\[ A = -2 \][/tex]
