Solve the problem using matrices.
-Find the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ for which $f ( 0 ) = - 2 , f ( 1 ) = 5 , f ( - 1 ) = 3$, $f ( 2 ) = 4$.

A) $f ( x ) = 10 x ^ { 3 } - 18 x ^ { 2 } - 13 x + 6$

B) $f ( x ) = - 8 x ^ { 3 } + 12 x ^ { 2 } + 5 x - 6$

C) $f ( x ) = - \frac { 10 } { 3 } x ^ { 3 } + 6 x ^ { 2 } + \frac { 13 } { 3 } x - 2$

D) $f ( x ) = \frac { 8 } { 3 } x ^ { 3 } + 4 x ^ { 2 } + \frac { 5 } { 3 } x - 2$

Question

Solve the problem using matrices.
-Find the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ for which $f ( 0 ) = - 2 , f ( 1 ) = 5 , f ( - 1 ) = 3$, $f ( 2 ) = 4$.

A) $f ( x ) = 10 x ^ { 3 } - 18 x ^ { 2 } - 13 x + 6$

B) $f ( x ) = - 8 x ^ { 3 } + 12 x ^ { 2 } + 5 x - 6$

C) $f ( x ) = - \frac { 10 } { 3 } x ^ { 3 } + 6 x ^ { 2 } + \frac { 13 } { 3 } x - 2$

D) $f ( x ) = \frac { 8 } { 3 } x ^ { 3 } + 4 x ^ { 2 } + \frac { 5 } { 3 } x - 2$

Quizplus · Accepted Answer

To find the coefficients $a$, $b$, $c$, and $d$, we set up the system of equations using the given conditions:  1. $d = -2$ from $f(0) = -2$.  2. $a + b + c + d = 5$ from $f(1) = 5$.  3. $-a + b - c + d = 3$ from $f(-1) = 3$.  4. $8a + 4b + 2c + d = 4$ from $f(2) = 4$.  Substituting $d = -2$ into the equations and solving the resulting system gives the coefficients:  $a = -\frac{10}{3}$, $b = 6$, $c = \frac{13}{3}$, $d = -2$.  Thus, the function is $f(x) = -\frac{10}{3}x^3 + 6x^2 + \frac{13}{3}x - 2$.

Solve the Problem Using Matrices f(x)=ax3+bx2+cx+df ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + df(x)=ax3+bx2+cx+d

Solve the Problem Using Matrices $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$