Use a system of equations to find the cubic function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ that satisfies given equations. Solve the system using matrices.
$$\begin{aligned}
f ( 2 ) & = - 26 \
f ( - 2 ) & = 6 \
f ( 1 ) & = 0 \
f ( - 1 ) & = - 2
\end{aligned}$$
A) $a = 2 , b = - 2 , c = 1 , d = - 1$
B) $a = - 26 , b = 6 , c = 0 , d = - 2$
C) $a = - 3 , b = - 3 , c = 4 , d = 2$
D) $a = - 26 , b = 30 , c = - 12 , d = 16$
E) $a = 50,598 , b = - 54,650 c = - 730 , d = 518$

Question

Quizplus · Accepted Answer

To solve for the coefficients $a$, $b$, $c$, and $d$ of the cubic function $f(x) = ax^3 + bx^2 + cx + d$, we substitute the given values of $x$ into the function and set them equal to the corresponding values of $f(x)$. This results in a system of linear equations:1. Substituting $x = 2$, we get $8a + 4b + 2c + d = -26$.2. Substituting $x = -2$, we get $-8a + 4b - 2c + d = 6$.3. Substituting $x = 1$, we get $a + b + c + d = 0$.4. Substituting $x = -1$, we get $-a + b - c + d = -2$.We can represent this system of equations as a matrix equation $AX = B$, where $A$ is the matrix of coefficients, $X$ is the column matrix of variables $[a, b, c, d]^T$, and $B$ is the column matrix of the constants on the right-hand side of the equations. Solving this matrix equation gives us the values of $a$, $b$, $c$, and $d$. The correct solution is $a = -3$, $b = -3$, $c = 4$, and $d = 2$, which corresponds to choice C.

Use a System of Equations to Find the Cubic Function f(x)=ax3+bx2+cx+df ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + df(x)=ax3+bx2+cx+d

Use a System of Equations to Find the Cubic Function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$