Question

Determine the equation of a cubic polynomial function with roots at (3,0), (2,0) and (-4,0) that also passes through the point (1,30).

Answer 1

Step-by-step explanation:

Step 1: Setting Up the Factors

Our roots are 3, 2, and -4 so

our factors are

$(x - 3)(x - 2)(x + 4)$

Step 2: Initial Test to see if the result equal 30

Let a be a constant, such we have

$a(x - 3)(x - 2)(x + 4) = 30$

Plug in. 1 for x.

$a(1 - 3)(1 - 2)(1 + 4) = 30$

$a( - 2)( - 1)(5) = 30$

$a(10) = 30$

$a = 3$

So our equation is

$3(x - 3)(x - 2)(x + 4)$

Or if you want it simplifed

$3( {x}^{2} - 5x + 6)(x + 4) = 3( {x}^{3} + - {x}^{2} - 14x + 24) = 3 {x}^{3} - 3 {x}^{2} - 42x + 72$

Answer 2

Answer:

$f(x)=3(x-3)(x-2)(x+4)$

Step-by-step explanation:

General form of a cubic polynomial function with 3 roots:

$f(x)=a(x-b)(x-c)(x-d)$

where:

• a is some constant to be found
• b, c and d are the roots of the function

Given roots:

• (3, 0)
• (2, 0)
• (-4, 0)

Substitute the given roots into the general form of the function:

$\implies f(x)=a(x-3)(x-2)(x-(-4))$

$\implies f(x)=a(x-3)(x-2)(x+4)$

To find the value of a, substitute the given point (1, 30) into the equation:

$\begin{aligned} f(1) & = 30\\\implies a(1-3)(1-2)(1+4) & =30\\a(-2)(-1)(5) & = 30 \\10a & = 30\\\implies a & = 3 \end{aligned}$

Therefore, the equation of the cubic polynomial function is:

$f(x)=3(x-3)(x-2)(x+4)$

Learn more about polynomials here:

brainly.com/question/27953978