To find the value of [tex]\( A \)[/tex] in the polynomial [tex]\( f(x) = 2x^3 + Ax^2 + 4x - 3 \)[/tex] given that [tex]\( f(2) = 13 \)[/tex], we can follow these steps:
1. Substitute [tex]\( x = 2 \)[/tex] into the polynomial [tex]\( f(x) \)[/tex]:
[tex]\[
f(2) = 2(2)^3 + A(2)^2 + 4(2) - 3
\][/tex]
2. Simplify each term:
[tex]\[
2(2)^3 = 2 \times 8 = 16
\][/tex]
[tex]\[
A(2)^2 = A \times 4 = 4A
\][/tex]
[tex]\[
4(2) = 8
\][/tex]
3. Combine terms:
[tex]\[
f(2) = 16 + 4A + 8 - 3
\][/tex]
4. Simplify the constants:
[tex]\[
16 + 8 - 3 = 21
\][/tex]
5. So, the equation is now:
[tex]\[
f(2) = 21 + 4A
\][/tex]
6. Given that [tex]\( f(2) = 13 \)[/tex]:
[tex]\[
21 + 4A = 13
\][/tex]
7. Solve for [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]\( -2 \)[/tex].
