Answered

Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

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).

Sagot :

culveryolanda46

Step-by-step explanation:

Step 1: Setting Up the Factors

Our roots are 3, 2, and -4 so

our factors are

[tex](x - 3)(x - 2)(x + 4)[/tex]

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

Let a be a constant, such we have

[tex]a(x - 3)(x - 2)(x + 4) = 30[/tex]

Plug in. 1 for x.

[tex]a(1 - 3)(1 - 2)(1 + 4) = 30[/tex]

[tex]a( - 2)( - 1)(5) = 30[/tex]

[tex]a(10) = 30[/tex]

[tex]a = 3[/tex]

So our equation is

[tex]3(x - 3)(x - 2)(x + 4)[/tex]

Or if you want it simplifed

[tex]3( {x}^{2} - 5x + 6)(x + 4) = 3( {x}^{3} + - {x}^{2} - 14x + 24) = 3 {x}^{3} - 3 {x}^{2} - 42x + 72[/tex]

View image culveryolanda46
semsee45

Answer:

[tex]f(x)=3(x-3)(x-2)(x+4)[/tex]

Step-by-step explanation:

General form of a cubic polynomial function with 3 roots:

[tex]f(x)=a(x-b)(x-c)(x-d)[/tex]

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:

[tex]\implies f(x)=a(x-3)(x-2)(x-(-4))[/tex]

[tex]\implies f(x)=a(x-3)(x-2)(x+4)[/tex]

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

[tex]\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}[/tex]

Therefore, the equation of the cubic polynomial function is:

[tex]f(x)=3(x-3)(x-2)(x+4)[/tex]

Learn more about polynomials here:

https://brainly.com/question/27953978

View image semsee45
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.