Question

Create an equation for a cubic function in standard form

Answer 1

f(x) = 2x³ - 10x² - 34x + 42

Explanation:

The standard form of a cubic equation is f(x) = ax³ + bx² + cx + d

Given conditions are:

First: Zeros of the cubic function are - 3, 1, and 7

Second: f(- 2) = 54

a(-3)³ + b(-3)² + c(-3) + d = 0 ⇔ - 27a + 9b - 3c + d = 0 ...... (1)

a + b + c + d = 0 ......... (2)

a(7)³ + b(7)² + c(7) + d = 0 ⇔ 343a + 49b + 7c + d = 0 ....... (3)

a(-2)³ + b(-2)² + c(-2) + d = 54 ⇔ - 8a + 4b - 2c + d = 54 ...... (4)

We have 4 equations with 4 unknown variables.

Use Cramer's rule to solve the system

A =
`\left[\begin{array}{cccc}-27&9&-3&1\\1&1&1&1\\343&49&7&1\\-8&2&-2&1\end{array}\right]` = 6,480

`A_(a)` =
`\left[\begin{array}{cccc}0&9&-3&1\\0&1&1&1\\0&49&7&1\\54&2&-2&1\end{array}\right]` = 12,960

`A_(b)` =
`\left[\begin{array}{cccc}-27&0&-3&1\\1&0&1&1\\343&0&7&1\\-8&54&-2&1\end{array}\right]` = - 64,800

`A_(c)` =
`\left[\begin{array}{cccc}-27&9&0&1\\1&1&0&1\\343&49&0&1\\-8&2&54&1\end{array}\right]` = - 220,320

`A_(d)` =
`\left[\begin{array}{cccc}-27&9&-3&0\\1&1&1&0\\343&49&7&0\\-8&2&-2&54\end{array}\right]` = 272,160

a =
`(A_(a) )/(A)` = 2

b =
`(A_(b) )/(A)` = - 10

c =
`(A_(c) )/(A)` = - 34

d =
`(A_(d) )/(A)` = 42

f(x) = 2x³ - 10x² - 34x + 42