Let y=f(x) be a cubic polynomial where f(-2) = f(1) = f(3) = 0, and f(0) = 20

Question

Determine f(x)

Question for Honors Precalculus

Answer 1

Hey! Lets solve this shall we ?

So, we must find out what f(x) is given f(-2) = f(1) = f(3) = 0 and f(0) = 20. This statement is saying that the zeros of the polynomial f are -2, 1 and 3 such that x = -2, x = 1 and x = 3. We must turn these zeros into their factors so we can find f, so

x = -2; (x + 2) = 0

x = 1; (x - 1) = 0

x = 3; (x - 3) = 0..........We must multiply these factors to obtain f such that

(x+2)(x-1)(x-3)...........Im gonna multiply the first two together first to get

(x² - x + 2x - 2)(x - 3)

(x² + x - 2)(x - 3)..........Then we multiply again such that

x³ + x² - 2x - 3x² - 3x + 6...........Then add like terms to obtain

f(x) = x³ - 2x² - 5x + 6

I hope this helped!

Answer 2

f(-2)=f(1)=f(3)=0 This tells you that these are the zeros. You can use the zeros to find the factors:

f(x)=a(x+2)(x-1)(x-3)

With this, you can use the last piece of information given to solve for a by plugging in (0,20)

20=a(0+2)(0-1)(0-3)

20=a(2)(-1)(-3)

20=6a Divide both sides by 6 and reduce

a=10/3

f(x)=10/3(x+2)(x-1)(x-3)

For the final answer, you will need to multiply the factors out and then distribute the 10/3 to all the terms.