Find a polynomial of the specified degree that has the given zeros:
Degree 4: zeros -1,1,3,5
(I know you would do...
(x-(-1)) (x-1) (x-3) (x-5) but I don't know what to do after that. I know when it's a degree of 3 you just use FOIL and leave the (x-5) as is, but how do you do it when there is 4?
u figure out the variable
To find a polynomial of degree 4 with the given zeros, you will need to multiply the factors (x-(-1)), (x-1), (x-3), and (x-5) together. Here's how you can do it:
Step 1: Rewrite the factors using positive coefficients:
(x+1)(x-1)(x-3)(x-5)
Step 2: Expand the product using the FOIL method for each pair of factors:
(x^2 - 1)(x-3)(x-5)
Step 3: Expand the product further:
[(x^2 - 1)(x-3)](x-5)
Step 4: Use the FOIL method again to find the product of (x^2 - 1) and (x-3):
[x^3 - 3x^2 - x + 3](x-5)
Step 5: Expand the final product:
x^4 - 5x^3 - 3x^2 + 15x^2 + 3x - 15
Step 6: Simplify the polynomial:
x^4 - 5x^3 + 12x^2 + 3x - 15
Therefore, the polynomial of degree 4 with zeros -1, 1, 3, and 5 is:
f(x) = x^4 - 5x^3 + 12x^2 + 3x - 15.